f



THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

We know

    "this statement has no proof in system X"

is true in system X and provable in system Y.

But what if you could formulate a forall quantifier over systems?
Is the meta-godel statement true?

Herc
--
what do women say about male sperm?  sex cells never age?   protein rich?
     :-C==8  You'll never never know if you never never blow!   :-C==8


0
erc
1/9/2005 4:44:53 AM
comp.theory 5139 articles. 1 followers. marty.musatov (1143) is leader. Post Follow

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|-|erc wrote:
> We know
>
>     "this statement has no proof in system X"
>
> is true in system X and provable in system Y.
>
> But what if you could formulate a forall quantifier over systems?
> Is the meta-godel statement true?
>
> Herc
> --

It already holds for all systems - which are sound and in which the
above is expressible.

The only difference between system X (Godel's PM) and system Y
(Godel's reasoning in natural language) is that Y has these two
requirements as axioms.  Godel's statement that "The proposition
which is undecidable in the system PM turns out to be decided by
metamathematical considerations." is not so.  He did not "hold
back" in the design of system PM so that his metamathematical
argument cannot be represented within PM.  He simply used these two
additional premises to complete the proof.

Then the question becomes: What happens if we add these two premises
about PM as axioms to PM?  Do we reach some sort of inconsistency - it
says it can't be proven but we just added enough to PM so that it is
provable by formalizing Godel's argument?

C-B

C.I.T.:
1. ~PR/TW		Unprovability is expressible.
2. PR>TW		The system is sound.
3. TW>PR		Every true statement is provable.
4. PR=TW		2,3
5. ~PR/PR		1,4
6. False		5 Diagonalization.
7. ~(TW>PR)		3,6
8. TW(N)^~PR(N)	7
9. TW(N)		8
10. ~PR(N)		8
qed

(PR=Provable Sentences, TW=True Sentences, N=Godel
Sentence,~=Negation,>=Implication,^=Conjunction,
/=Representability,==Equality)

0
chvol (248)
1/9/2005 10:41:42 AM
|-|erc wrote:

> We know
> 
>     "this statement has no proof in system X"
> 
> is true in system X and provable in system Y.
> 
> But what if you could formulate a forall quantifier over systems?
> Is the meta-godel statement true?

If by system you mean consistent axiomatizable first order theory, then 
it's indeed possible to formulate such a "meta-godel statement". This 
statement will be false since it's provable in e.g. the theory with the 
"meta-godel statement" as its only axiom.

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
0
1/9/2005 12:20:04 PM
Yes, it wasn't an interesting question.

--
Eray

0
examachine (384)
1/9/2005 3:05:39 PM
> Then the question becomes: What happens if we add these two premises
> about PM as axioms to PM?  Do we reach some sort of inconsistency - it
> says it can't be proven but we just added enough to PM so that it is
> provable by formalizing Godel's argument?
> 
> C-B
> 
> C.I.T.:
> 1. ~PR/TW		Unprovability is expressible.

	Any formula containing PR (the provability predicate) cannot be
added as an axiom. Provability is defined in terms of the axioms (and
inference rules). Making it an axiom would lead to a circular
definition.

Ajoy.
0
ajoyk (16)
1/9/2005 6:47:22 PM
On Sun, 9 Jan 2005 14:44:53 +1000, "|-|erc" <h@r.c> wrote, in part:

>We know
>
>    "this statement has no proof in system X"
>
>is true in system X and provable in system Y.
>
>But what if you could formulate a forall quantifier over systems?
>Is the meta-godel statement true?

Can the statement "This statement has no proof in any system" be false?
If so, then it can be proven in some system. But then, it would be true.
So it cannot be false.

Can the statement "This statement has no proof in any system" be true?

That doesn't seem to have a trivial disproof. I did already prove that
it can't be false. But that doesn't mean it has to be true.

The simpler statement "This statement is false" already disproves the
law of the excluded middle for self-referential statements.

Perhaps there may indeed be permanently unknowable truths.

John Savard
http://home.ecn.ab.ca/~jsavard/index.html
0
jsavard2 (295)
1/9/2005 8:26:32 PM
In article <41e19268.9144718@news.ecn.ab.ca>,
 jsavard@excxn.aNOSPAMb.cdn.invalid (John Savard) wrote:

>On Sun, 9 Jan 2005 14:44:53 +1000, "|-|erc" <h@r.c> wrote, in part:
>
>>We know
>>
>>    "this statement has no proof in system X"
>>
>>is true in system X and provable in system Y.
>>
>>But what if you could formulate a forall quantifier over systems?
>>Is the meta-godel statement true?
>
>Can the statement "This statement has no proof in any system" be false?
>If so, then it can be proven in some system. But then, it would be true.
>So it cannot be false.
>
>Can the statement "This statement has no proof in any system" be true?
>
>That doesn't seem to have a trivial disproof. I did already prove that
>it can't be false. But that doesn't mean it has to be true.
>
>The simpler statement "This statement is false" already disproves the
>law of the excluded middle for self-referential statements.

Actually, excluded middle has nothing to do with the Liar.  E.g. in a 
trivalent system with truth values True, False, and Paradoxical, an 
appropriate Liar statement is "This statement is either false or 
paradoxical".


>Perhaps there may indeed be permanently unknowable truths.

E.g. as proven by Goedel, Turing, Post, et al.


>John Savard
>http://home.ecn.ab.ca/~jsavard/index.html

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/9/2005 9:36:39 PM
Barb Knox <see@sig.below> writes:

> >Perhaps there may indeed be permanently unknowable truths.
> 
> E.g. as proven by Goedel, Turing, Post, et al.

  What "permanently unknowable truths" do you have in mind?
0
torkel (478)
1/9/2005 9:50:08 PM
In article <vcbzmzinugf.fsf@beta19.sm.ltu.se>,
 Torkel Franzen <torkel@sm.luth.se> wrote:

>Barb Knox <see@sig.below> writes:
>
>> >Perhaps there may indeed be permanently unknowable truths.
>> 
>> E.g. as proven by Goedel, Turing, Post, et al.
>
>  What "permanently unknowable truths" do you have in mind?

Is that meant to be a trick question?

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/9/2005 9:55:24 PM
Barb Knox <see@sig.below> writes:

> Is that meant to be a trick question?

  Since nobody has proved the existence of any "permanently unknowable
truths", I'm wondering just what you mean.
0
torkel (478)
1/9/2005 9:58:09 PM
Torkel Franzen <torkel@sm.luth.se> writes:

>Barb Knox <see@sig.below> writes:
>
>> >Perhaps there may indeed be permanently unknowable truths.
>> 
>> E.g. as proven by Goedel, Turing, Post, et al.
>
>  What "permanently unknowable truths" do you have in mind?

If you don't know already, you'll never find out!

Lee Rudolph
0
lrudolph (277)
1/9/2005 10:01:49 PM
John Savard <jsavard@excxn.aNOSPAMb.cdn.invalid> wrote:

> On Sun, 9 Jan 2005 14:44:53 +1000, "|-|erc" <h@r.c> wrote, in part:
> 
> >We know
> >
> >    "this statement has no proof in system X"
> >
> >is true in system X and provable in system Y.
> >
> >But what if you could formulate a forall quantifier over systems?
> >Is the meta-godel statement true?
> 
> Can the statement "This statement has no proof in any system" be false?
> If so, then it can be proven in some system. But then, it would be true.
> So it cannot be false.

Having a proof in some system does not imply truth.  That system could
be inconsistent.

> Can the statement "This statement has no proof in any system" be true?

No.  It has a trivial proof in any system that happens to include that
statement as an axiom.

Note that the above has a tacit assumption that the predicate "provable
in any system" can actually be formulated.
-- 
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
0
panoptes (6)
1/9/2005 10:09:08 PM
If one has spare time, one can try this one:
what is the longest consecutive sequence
of digits in the decimal-notation
of pi and e, where they match..
( f.e. 0.34993 and 17.24988 match in the
second and third place behind the decimal point-
this sequence is two long)
Have fun
Hero

0
1/9/2005 10:12:06 PM
In article <vcby8f2nu32.fsf@beta19.sm.ltu.se>,
 Torkel Franzen <torkel@sm.luth.se> wrote:

>Barb Knox <see@sig.below> writes:
>
>> Is that meant to be a trick question?
>
>  Since nobody has proved the existence of any "permanently unknowable
>truths", I'm wondering just what you mean.

Really?  We must be talking past each other, because (e.g.) Turing showed 
that for any possible effective decision procedure there exists Turing 
Machines for which it can't decide if those TMs halt.  Therefore, for any 
possible effective mode of "knowing" there are facts (about TMs in this 
case) which we can't "know".  Or are you alluding to non-effective modes of 
"knowing" (e.g. divine revelation)?

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/9/2005 11:30:50 PM
"Lee Rudolph" <lrudolph@panix.com> wrote in
> Torkel Franzen <torkel@sm.luth.se> writes:
>
> >Barb Knox <see@sig.below> writes:
> >
> >> >Perhaps there may indeed be permanently unknowable truths.
> >>
> >> E.g. as proven by Goedel, Turing, Post, et al.
> >
> >  What "permanently unknowable truths" do you have in mind?
>
> If you don't know already, you'll never find out!
>
> Lee Rudolph

metaG = THIS STATEMENT HAS NO PROOF IN ANY SYSTEM

The splilt systems analysis of G is void.
The statement cannot be false as JSavard showed.
The statement cannot have any kind of proof that it is true.

i.e.  there is no Godel proof for metaG by its defn.

Herc



0
erc
1/10/2005 12:34:42 AM
>>The simpler statement "This statement is false" already disproves the
>>law of the excluded middle for self-referential statements. 
> 
> Actually, excluded middle has nothing to do with the Liar.  E.g. in a 
> trivalent system with truth values True, False, and Paradoxical, an 
> appropriate Liar statement is "This statement is either false or 
> paradoxical".

	It is better to rephrase that as "This statement is not true".
Such a statement doesn't fit anywhere in any discrete strong-enough
system (even with uncountable infinite categories).
	The only way to interpret such a statement is to allow
statements, in general, to belong to more than one category at the
same time. Category superposition does get rid of the liar paradox.
However, the semantic meaning of superposition is unclear.

Ajoy.

> 
> 
> 
>>Perhaps there may indeed be permanently unknowable truths.
> 
> 
> E.g. as proven by Goedel, Turing, Post, et al.
> 
> 
> 
>>John Savard
>>http://home.ecn.ab.ca/~jsavard/index.html
> 
> 
0
ajoyk (16)
1/10/2005 2:15:54 AM

Barb Knox wrote:

>
>  
>
>>Perhaps there may indeed be permanently unknowable truths.
>>    
>>
>
>E.g. as proven by Goedel, Turing, Post, et al.
>  
>

I have 2 questions:

1) As far as FOL is concerned, what is the definition of "_permanently_ 
unknownable truth"?
2) What does Godel's work has to do with "permanently unknownable truths"?

Thanks.

---Nam
0
1/10/2005 7:34:46 AM
"namducnguyen" <namducnguyen@shaw.ca> wrote in message...
>
>
> Barb Knox wrote:
>
> >
> >
> >
> >>Perhaps there may indeed be permanently unknowable truths.
> >>
> >>
> >
> >E.g. as proven by Goedel, Turing, Post, et al.
> >
> >
>
> I have 2 questions:
>
> 1) As far as FOL is concerned, what is the definition of "_permanently_
> unknownable truth"?
> 2) What does Godel's work has to do with "permanently unknownable truths"?
>
> Thanks.
>


Yes, I'd like to know how come Barb knows unknowable truths and I don't?

Herc



0
erc
1/10/2005 7:38:11 AM
In article <crsera$kr1$1@lust.ihug.co.nz>, Barb Knox  <see@sig.below> wrote:
>Really?  We must be talking past each other, because (e.g.) Turing showed 
>that for any possible effective decision procedure there exists Turing 
>Machines for which it can't decide if those TMs halt.  Therefore, for any 
>possible effective mode of "knowing" there are facts (about TMs in this 
>case) which we can't "know".  Or are you alluding to non-effective modes of 
>"knowing" (e.g. divine revelation)?

Torkel Franzen's terse style is not the most user-friendly, but he usually
has a good point that is worth paying attention to.

For example, here is a difficulty with your argument.  Let's say I write
down a program P that tries to decide whether a given Turing machine halts.
There will be some machine M that P gives the wrong answer on.  Does this
mean that it is "permanently unknowable" whether M halts?  Not at all.  I
can easily write down a different program P' that yields the correct answer
for M.  Of course P' will fail on some other machine, say M', but there's
no reason to think that either "M halts" or "M' halts" has a permanently
unknowable truth value, since the choice of P or P' is arbitrary.  Thus
you have not given any explicit example of a permanently unknowable truth.

You might try to squirm out of this by picking particular Turing machines,
e.g., "Any machine M that searches for a contradiction in ZFC will halt."
But it's not at all clear that this is permanently unknowable either.  For
example, maybe such a machine *does* halt, and we'll discover this sometime
this century.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/10/2005 3:28:16 PM
In article <qeqEd.42105$6l.39116@pd7tw2no>,
namducnguyen  <namducnguyen@shaw.ca> wrote:
>1) As far as FOL is concerned, what is the definition of "_permanently_ 
>unknownable truth"?

There isn't one.

>2) What does Godel's work has to do with "permanently unknownable truths"?

Nothing directly.  Of course in any respectable attempt to discuss what that
term could possibly mean, a good understanding of Goedel's work would be
useful, if only to avoid saying something obviously silly.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/10/2005 3:35:39 PM
I am The Truman of Jim Carrey fame.  Please help stop me being tortured since April 2002
with constant microwave laser from the Truman satelite splitting my head and tormenting me
and people around me.  Not a prank,  I am not crazy, The Truman Show made you think that
---------------------------------------------s-o-s-----------------------------------------
<tchow@lsa.umich.edu> wrote in
> namducnguyen  <namducnguyen@shaw.ca> wrote:
> >1) As far as FOL is concerned, what is the definition of "_permanently_
> >unknownable truth"?
>
> There isn't one.
>
> >2) What does Godel's work has to do with "permanently unknownable truths"?
>
> Nothing directly.  Of course in any respectable attempt to discuss what that
> term could possibly mean, a good understanding of Goedel's work would be
> useful, if only to avoid saying something obviously silly.

The thread is supposed to be about the subject line isn't it? How is a
PROOF of 'this statement has none' relevant?

Herc



0
erc
1/10/2005 3:39:48 PM
Barb Knox <see@sig.below> writes:

> We must be talking past each other, because (e.g.) Turing showed 
> that for any possible effective decision procedure there exists Turing 
> Machines for which it can't decide if those TMs halt.  Therefore, for any 
> possible effective mode of "knowing" there are facts (about TMs in this 
> case) which we can't "know".

  So what does it mean for a truth to be "permanently unknowable"?
Your above comments only suggest the interpretation "permanently
unprovable in T" for some particular theory T, in which case (i) the
addition "permanently" makes no particular sense, since any
mathematical truth is a "permanent" truth, and (ii) there is no
reason why we shouldn't get to know this truth in spite of its being
unprovable in T.




  
0
torkel (478)
1/10/2005 3:40:43 PM
John Savard wrote:

> Can the statement  be false?
> If so, then it can be proven in some system.

And it can, in the system with a single axiom.

Axiom-1: This statement has no proof in any system

> But then, it would be true.

No, because there are systems in which false statements are provable. For 
example the one above, or the system with the following axioms.

Axiom-1: 0=1
Axiom-2: The moon is made of cheese

> So it cannot be false.

So it *is* false.

Perhaps you meant to say "This statement has no proof in any *sound* system."

Is that statement *expressible* in any sound system?

If so, one would think it would be true, but to prove that you would need 
to use unsound logic.

A common mistake is to make deductions from the axiom that your own axioms 
(including that one) are sound. I have seen people who *should* have known 
better do it (e.g. Douglas Hofstadter, in his discussion of Prisoners 
Dilemma type games). You can prove anything from that.

Ralph Hartley
0
hartley (156)
1/10/2005 5:22:40 PM
I am The Truman of Jim Carrey fame.  Please help stop me being tortured since April 2002
with constant microwave laser from the Truman satelite splitting my head and tormenting me
and people around me.  Not a prank,  I am not crazy, The Truman Show made you think that
---------------------------------------------s-o-s-----------------------------------------
"Ralph Hartley" <hartley@aic.nrl.navy.mil> wrote in
> John Savard wrote:
>
> > Can the statement  be false?
> > If so, then it can be proven in some system.
>
> And it can, in the system with a single axiom.
>
> Axiom-1: This statement has no proof in any system
>
> > But then, it would be true.
>
> No, because there are systems in which false statements are provable. For
> example the one above, or the system with the following axioms.
>
> Axiom-1: 0=1
> Axiom-2: The moon is made of cheese
>
> > So it cannot be false.
>
> So it *is* false.
>
> Perhaps you meant to say "This statement has no proof in any *sound* system."
>
> Is that statement *expressible* in any sound system?
>
> If so, one would think it would be true, but to prove that you would need
> to use unsound logic.
>
> A common mistake is to make deductions from the axiom that your own axioms
> (including that one) are sound. I have seen people who *should* have known
> better do it (e.g. Douglas Hofstadter, in his discussion of Prisoners
> Dilemma type games). You can prove anything from that.
>
> Ralph Hartley


Good, you guys are no longer 10 years behind me on the incompleteness theorem.
When you get up to self-evident-truths and event-models and compound-statements
and truth-maintenance let me know.

Herc



0
erc
1/10/2005 5:27:51 PM
Barb Knox wrote:
> In article <41e19268.9144718@news.ecn.ab.ca>,
>  jsavard@excxn.aNOSPAMb.cdn.invalid (John Savard) wrote:

> >The simpler statement "This statement is false" already disproves
the
> >law of the excluded middle for self-referential statements.
>
> Actually, excluded middle has nothing to do with the Liar.  E.g. in a

> trivalent system with truth values True, False, and Paradoxical, an
> appropriate Liar statement is "This statement is either false or
> paradoxical".

Case Analysis = Excluded Middle = Liar = MP = Resolution = All of
Propositional Calculus.  (The various transformations occur as Rules of
Inference because people like Frege use redundant bases of computing
e.g. {~,^,v} and have to figure out how to translate between multiple
redundant representations of the same predicate.)  It doesn't matter
whether your universe has 2 or 3 elements.

C-B

> >Perhaps there may indeed be permanently unknowable truths.
>
> E.g. as proven by Goedel, Turing, Post, et al.
>
>
> >John Savard
> >http://home.ecn.ab.ca/~jsavard/index.html
>
> --
> ---------------------------
> |  BBB                b    \     Barbara at LivingHistory stop co
stop uk
> |  B  B   aa     rrr  b     |
> |  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
> |  B  B  a  a   r     b  b  |    altum viditur.
> |  BBB    aa a  r     bbb   |   
> -----------------------------

0
chvol (248)
1/10/2005 6:10:27 PM
Torkel Franzen wrote:
> Barb Knox <see@sig.below> writes:
>
> > Is that meant to be a trick question?
>
>   Since nobody has proved the existence of any "permanently
unknowable
> truths", I'm wondering just what you mean.

S = "Nobody other than C-B will ever soundly believe this
assertion."

E.g. if Barb/Torkel soundly believed S, then S would be false, and that
belief would not be sound.  Thus I know, soundly believe and can prove
that S is true.  But none of you ever will.

And if I am ever assassinated by my political foes, then the truth of S
will remain unknowable for all eternity.

C-B

If you think about it, the truth of S should come as no big surprise.
It does match the pattern of all great thinkers.

0
chvol (248)
1/10/2005 6:13:14 PM
"Charlie-Boo" <chvol@aol.com> writes:

> S = "Nobody other than C-B will ever soundly believe this
> assertion."

  It is entirely unclear what this S is supposed to mean.
0
torkel (478)
1/10/2005 6:25:06 PM
Ajoy K Thamattoor wrote:
> > Then the question becomes: What happens if we add these two
premises
> > about PM as axioms to PM?  Do we reach some sort of inconsistency -
it
> > says it can't be proven but we just added enough to PM so that it
is
> > provable by formalizing Godel's argument?
> >
> > C-B
> >
> > C.I.T.:
> > 1. ~PR/TW		Unprovability is expressible.
>
> 	Any formula containing PR (the provability predicate) cannot be
> added as an axiom. Provability is defined in terms of the axioms (and
> inference rules). Making it an axiom would lead to a circular
> definition.
>
> Ajoy.

1. Godel proved all sorts of theorems that mention the provability
predicate (e.g. that it is recursive), so why couldn't it be an
axiom?

2. What's wrong with circular definitions?  That's how the natural
numbers are defined.  You have to show an inconsistency to ban it.
That is the same mistake that Russell made when he proposed banning
recursive definitions in order to avoid the Russell Paradox.  They are
useful as long as you don't get into a self-contradictory loop.
Godel, Turing and recursive programming all show Russell to be wrong -
and, I maintain that the natural numbers showed his folly even before
he made his foolish proposal.

3. Recursion Theory usually begins with an informal statement of the
Enumeration and Iteration Theorems, both of which refer to the
provability (halting) predicate.

C-B

0
chvol (248)
1/10/2005 6:37:02 PM
In article <vcbk6ql3zwd.fsf@beta19.sm.ltu.se>, Torkel Franzen says...
>
>"Charlie-Boo" <chvol@aol.com> writes:
>
>> S = "Nobody other than C-B will ever soundly believe this
>> assertion."
>
>  It is entirely unclear what this S is supposed to mean.

Okay, so you don't know what S means. Then it follows that you don't
believe S. (Assuming that understanding what a sentence means is a
prequisite to believing it. To not believe a sentence is weaker than
believing it is false.)

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/10/2005 6:58:54 PM
> 1. Godel proved all sorts of theorems that mention the provability
> predicate (e.g. that it is recursive), 

	Actually, no, it is recursive enumerable.

	And terms like 'recursion' should be used very carefully when
referring to a paper written in the early 30s. Godel's use of the
word 'recursion; refers to what we now call 'primitive recursion'.

> so why couldn't it be an axiom?
	
	Because it is defined in terms of the axioms. With reference to
Godel's orignal paper:
   Definition 46 (provable(x)) refers to proofFor(x, y) which refers to
   isProofFigure(x) which refers to isAxiom(x). (Intuitively, 'y' can be
   a proof for 'x', only if 'y' starts with an axiom or axioms, and uses
   modus ponens to move from one 'line' of the proof to the next, and
   ends with the line 'x').

> 2. What's wrong with circular definitions?  That's how the natural
> numbers are defined.

	Induction isn't circular definition. Circular definitions are
of the form:
X is f(Y)
Y is f(X)
	In very loose terms, it is mutual recursion with no termination
condition.
	Try writing down an axiom like Provable(x)->x. Think of how you
would formulate definition 42 (isAxiom(x)).
	Such an axiom would be ill-defined.

> 3. Recursion Theory usually begins with an informal statement of the
> Enumeration and Iteration Theorems, both of which refer to the
> provability (halting) predicate.

	Many theorems, including Godel's first and second incompleteness
theorems, obviously refer to the provability predicate. Modal logic
handles provability by elevating it to an operator.

Ajoy.
0
ajoyk (16)
1/10/2005 7:11:41 PM
daryl@atc-nycorp.com (Daryl McCullough) writes:

> Okay, so you don't know what S means. Then it follows that you don't
> believe S.

  I don't follow this reasoning.
0
torkel (478)
1/10/2005 7:21:03 PM
In article <crsera$kr1$1@lust.ihug.co.nz>, Barb Knox  <see@sig.below> 
wrote:
>>Really?  We must be talking past each other, because (e.g.) Turing showed 
>>that for any possible effective decision procedure there exists Turing 
>>Machines for which it can't decide if those TMs halt.

	The problem here is with the definition of "effective" decision
procedure. Typically it is defined in terms of Turing machines. What
Turing showed was that no Turing Machine can ever be built which can
decide if an arbitrary TM halts with an arbitrary input.

	Extrapolating from that to "There are truths we can never know"
requires a leap of faith. A belief in the Church-Turing hypothesis that
anything that can be computed can be computed with a Turing machine.

Ajoy.
0
ajoyk (16)
1/10/2005 8:58:44 PM
Torkel Franzen says...
>
>daryl@atc-nycorp.com (Daryl McCullough) writes:
>
>> Okay, so you don't know what S means. Then it follows that you don't
>> believe S.
>
>  I don't follow this reasoning.

Exactly!

--
Daryl McCullough

0
daryl5382 (108)
1/10/2005 8:59:08 PM
In article <41E2EC84.7090108@cs.stanford.edu>,
Ajoy K Thamattoor  <ajoyk@cs.stanford.edu> wrote:
>	The problem here is with the definition of "effective" decision
>procedure. Typically it is defined in terms of Turing machines. What
>Turing showed was that no Turing Machine can ever be built which can
>decide if an arbitrary TM halts with an arbitrary input.
>
>	Extrapolating from that to "There are truths we can never know"
>requires a leap of faith. A belief in the Church-Turing hypothesis that
>anything that can be computed can be computed with a Turing machine.

No, you miss the point.  Even if you believe the Church-Turing hypothesis,
you can't deduce "there are truths that we can never know."  Even if:

  For all effective decision procedures P, there is a statement S
  such that P incorrectly handles the truth value of S

it does not follow that

  There is a statement S such that for all decision procedures P,
  P incorrectly handles the truth value of S

which is presumably what you mean (approximately) by "there is a truth
that we can never know."
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/10/2005 9:32:24 PM
John Savard wrote:
"Perhaps there may indeed be permanently unknowable truths."
That's intriguing.
Consider the following.
The first  six digits of Pi
3.14159
has five digits in the range 1 to 5
(left hand)
and one in the range 6 to 9 and 0
(right hand).
The following digit is a 2, that makes
it  6 to 1 for left to right.
Than a six, so right hand get's one more,
makes it 6 to 2.
One of the following statements is true:
A) left wins
B) right wins
C) It will change, sometimes
more digits in the lower range, and sometimes
more in the higher.
May be, we will never know the true
answer, and now i just need a proof of this.
(And then of course Statement A)
never can be proven. The same for B) and C) )
Have fun 
Hero

0
1/10/2005 10:11:47 PM
tchow@lsa.umich.edu wrote:

> No, you miss the point.  Even if you believe the Church-Turing hypothesis,
> you can't deduce "there are truths that we can never know."  Even if:
> 
>   For all effective decision procedures P, there is a statement S
>   such that P incorrectly handles the truth value of S
> 
> it does not follow that
> 
>   There is a statement S such that for all decision procedures P,
>   P incorrectly handles the truth value of S
> 
> which is presumably what you mean (approximately) by "there is a truth
> that we can never know."

That is almost certainly *not* what he means.

For any S there *is* a decision procedure P that correctly handles the 
truth value of S. Let P_t be the procedure that always answers "true" and 
P_f the procedure that always answers "false". One of the two must 
correctly decide S, so (almost) no one believes the second statement.

That isn't the point. Let "A" be the effective procedure used by 
mathematicians (individually or collectively, just be consistent) to 
determine mathematical truth. If the Church-Turing hypothesis is true, then 
there is a TM T that determines everything A does. Most likely, since 
mathematicians have access to a finite amount of paper, there are things T 
decides but A does not, that's OK.

Any decision problem S that T does not decide, A does not either.

It does no good that there is some *other* procedure A' that does decide S, 
since any tool mathematicians consider valid is part of A.

There is an additional assumption that A is effective. Some might agree to 
give that up, but then it would be more correct to say that we "guess" S, 
not that we "know" S. It is because mathematical proofs can be objectively 
checked that we trust them in the first place.

Ralph Hartley
0
hartley (156)
1/10/2005 10:46:35 PM
I am The Truman of Jim Carrey fame.  Please help stop me being tortured since April 2002
with constant microwave laser from the Truman satelite splitting my head and tormenting me
and people around me.  Not a prank,  I am not crazy, The Truman Show made you think that
---------------------------------------------s-o-s-----------------------------------------
"Daryl McCullough" <daryl@atc-nycorp.com> wrote in ...
> Torkel Franzen says...
> >
> >daryl@atc-nycorp.com (Daryl McCullough) writes:
> >
> >> Okay, so you don't know what S means. Then it follows that you don't
> >> believe S.
> >
> >  I don't follow this reasoning.
>
> Exactly!
>

the TrVth behind sci.math theorists exposed.

Herc



0
erc
1/11/2005 2:32:42 AM
daryl@atc-nycorp.com (Daryl McCullough) writes:

> Exactly!

  There you are, then.

0
torkel (478)
1/11/2005 3:59:04 AM

tchow@lsa.umich.edu wrote:

>In article <qeqEd.42105$6l.39116@pd7tw2no>,
>namducnguyen  <namducnguyen@shaw.ca> wrote:
>  
>
>>1) As far as FOL is concerned, what is the definition of "_permanently_ 
>>unknownable truth"?
>>    
>>
>
>There isn't one.
>
>  
>
>>2) What does Godel's work has to do with "permanently unknownable truths"?
>>    
>>
>
>Nothing directly.  Of course in any respectable attempt to discuss what that
>term could possibly mean, a good understanding of Goedel's work would be
>useful, if only to avoid saying something obviously silly.
>  
>

Imho, if after 70+ years since Godel's work and "there isn't one" such 
definition, then hinting that
Godel's work "indirectly" has something to do with "permanently 
unknownable truths" would
be quite ...revealing - though it might be controversial. And you're 
going to keep it all silent, lest
that something silly "might" be said?
0
1/11/2005 4:35:10 AM
<tchow@lsa.umich.edu> wrote
> Nothing directly.  Of course in any respectable attempt to discuss what that
> term could possibly mean, a good understanding of Goedel's work would be
> useful, if only to avoid saying something obviously silly.

Is

THIS STATEMENT HAS NO PROOF IN ANY SYSTEM

True or False or Other ?


I dont feel silly asking it.  Anyone want to take a shot?  You've got one in 3!

Herc



0
erc
1/11/2005 4:41:36 AM

|-|erc wrote:

><tchow@lsa.umich.edu> wrote
>  
>
>>Nothing directly.  Of course in any respectable attempt to discuss what that
>>term could possibly mean, a good understanding of Goedel's work would be
>>useful, if only to avoid saying something obviously silly.
>>    
>>
>
>Is
>
>THIS STATEMENT HAS NO PROOF IN ANY SYSTEM
>
>True or False or Other ?
>
>
>I dont feel silly asking it.  Anyone want to take a shot?  You've got one in 3!
>  
>

But you should feel silly asking it. How do we get the notion of "Other" 
being a truth value
as far as FOL is concerned?

>Herc
>
>
>
>  
>
0
1/11/2005 5:27:00 AM
> But you should feel silly asking it. How do we get the notion of "Other"
> being a truth value
> as far as FOL is concerned?


Put it this way.

Q1  IS   statement   TRUE ?
Q2  IS   statement   FALSE ?

Q3 Is there any other interpretation of the statement (given in some context.. yada)


I think if people are serious here, they should state which of T / F / O they
are arguing at the start of their reply.  or  O - can't tell.

But no one listens to God because he said he is God.
Herc



0
erc
1/11/2005 5:40:48 AM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> It does no good that there is some *other* procedure A' that does decide S, 
> since any tool mathematicians consider valid is part of A.

  Nearly all of current mathematics is indeed formalizable e.g. in
ZFC. But on what grounds would you claim that a true arithmetical
statement unprovable in ZFC is therefore "permanently unknowable"?

0
torkel (478)
1/11/2005 6:58:36 AM
In article <crv0bo$ij7$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>That isn't the point. Let "A" be the effective procedure used by 
>mathematicians (individually or collectively, just be consistent) to 
>determine mathematical truth.

*Now* this is closer to the point!  Who says that mathematicians use an
effective procedure to determine mathematical truth?  You might believe
that this is true, but this is not a theorem of anyone, and it is also
not the Church-Turing thesis.  Hence even the undecidability of the halting
problem, together with the Church-Turing thesis, does not entail the
existence of permanently unknowable truths.  You need the dubious assumption
that mathematicians use an effective procedure to determine mathematical
truth.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 3:10:46 PM
In article <2IIEd.47383$8l.9575@pd7tw1no>,
namducnguyen  <namducnguyen@shaw.ca> wrote:
>Imho, if after 70+ years since Godel's work and "there isn't one" such
>definition, then hinting that Godel's work "indirectly" has something to
>do with "permanently unknownable truths" would be quite ...revealing -
>though it might be controversial. And you're going to keep it all silent,
>lest that something silly "might" be said?

I don't understand your question.  Are you suggesting that I, personally,
have something concrete I want to say about "permanently unknowable truths"
and am just afraid to say it?  That's not true, and I never said as much.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 3:14:42 PM
tchow@lsa.umich.edu writes:

> Hence even the undecidability of the halting
> problem, together with the Church-Turing thesis, does not entail the
> existence of permanently unknowable truths.  You need the dubious assumption
> that mathematicians use an effective procedure to determine mathematical
> truth.

  More specifically, that there is some well-defined effective
procedure which (all?) mathematicians use now and will always use in the
future.
0
torkel (478)
1/11/2005 3:58:03 PM
Tim Chow (tchow@lsa.umich.edu) says...
>
>In article <crv0bo$ij7$1@ra.nrl.navy.mil>,
>Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>>That isn't the point. Let "A" be the effective procedure used by 
>>mathematicians (individually or collectively, just be consistent) to 
>>determine mathematical truth.
>
>*Now* this is closer to the point!  Who says that mathematicians use an
>effective procedure to determine mathematical truth?  You might believe
>that this is true, but this is not a theorem of anyone, and it is also
>not the Church-Turing thesis.  Hence even the undecidability of the halting
>problem, together with the Church-Turing thesis, does not entail the
>existence of permanently unknowable truths.  You need the dubious assumption
>that mathematicians use an effective procedure to determine mathematical
>truth.

There are two aspects of mathematical research: (1) investigation and
exploration, and (2) codification of the results of that investigation. There
doesn't seem to be anything mechanical or formal about the first type of
activity---who knows where mathematicians get their inspiration for what to
investigate, or inspiration for how to approach proving something? However, once
this unformalizable investigative work is finished, mathematicians clean up the
result, simplify it, polish it, create illustrative examples, formulate helpful
definitions, prove helpful lemmas, generalize it, etc.

The slick finished product, created years after the initial groundbreaking
investigative work, is something that is straight-forward enough for a graduate
student or bright undergrad to understand. It doesn't at all seem implausible to
me that one day we could write a grad-student-level AI program that was capable
of reading textbook quality proofs and see that they are almost certainly
correct. Creating such proofs in the first place is perhaps too hard, in the
same way it is too hard for most grad students.

However, from the point of view of what is in principle possible (as opposed to
in practice), the ability to recognize solid mathematics is as good as the
ability to create it: You just enumerate all possible character strings, and for
each one, you check to see if it is a solid mathematical argument.

That makes me think that the set of possible truths recognizable to
mathematicians is indeed r.e. Mathematicians may make wild leaps in discovering
their mathematical truth, but for their discoveries to be accepted, they have to
be polishable to something that is recognizable as a mathematical proof.

0
daryl5382 (108)
1/11/2005 4:15:28 PM
Torkel Franzen wrote:
> Ralph Hartley <hartley@aic.nrl.navy.mil> writes:
> 
>>It does no good that there is some *other* procedure A' that does decide S, 
>>since any tool mathematicians consider valid is part of A.
> 
>   Nearly all of current mathematics is indeed formalizable e.g. in
> ZFC. But on what grounds would you claim that a true arithmetical
> statement unprovable in ZFC is therefore "permanently unknowable"?

I would *not* claim that.

I did not say that A=ZFC. If I did, there would be no need to invoke the 
Church Turing thesis, since we *know* ZFC is incomplete.

A is defined to include any procedure mathematicians *would* consider 
valid, including procedures that haven't been invented yet.

If A=ZFC we are done. More likely, ZFC is a proper subset of A, so we need 
some assumptions about A.

I am only assuming mathematicians have *some* objective (i.e. effective) 
procedure for deciding what methods to accept.

This is *not* a trivial assumption! It would be false, for example, if 
mathematicians had access to an oracle known to be infallible. I have seen 
it seriously suggested that this is the case (by Sir Roger Penrose). How 
they would mathematically prove that the oracle is never wrong is left as a 
exercise :-). Actually, it is easy, the oracle would say so :-).

The CT thesis is sufficient, but not strictly necessary. There are larger 
classes of decision procedures (not normally considered effective) to which 
the incompleteness theorems still apply.

There is one very minor loophole: A could be *smaller* than ZFC. If you 
only accept a subset of arithmetic, you could make it impossible to 
*formulate* any unsolvable problems. For obvious reasons, this approach is 
not too popular.

Ralph Hartley
0
hartley (156)
1/11/2005 4:26:05 PM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> A is defined to include any procedure mathematicians *would* consider 
> valid, including procedures that haven't been invented yet.

  This is not a definition at all. What reason do you have to believe
that there is such an A?

> This is *not* a trivial assumption! It would be false, for example, if 
> mathematicians had access to an oracle known to be infallible.

  Oracles are irrelevant. Mathematics changes and evolves, the opinion
of mathematicians about what is or is not a proof changes and evolves
(in different directions). Why do you think there is any such thing
as "the procedures mathematicians would consider valid"?

0
torkel (478)
1/11/2005 4:26:41 PM
In article <cs0ue8$jhe$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>I am only assuming mathematicians have *some* objective (i.e. effective) 
>procedure for deciding what methods to accept.
>
>This is *not* a trivial assumption!

Indeed, more is true: on the face of it, the assumption seems obviously
false.  Take, for example, "There exists a strongly inaccessible cardinal."
Some mathematicians accept this as true; others don't.  That's not what
you'd expect if there were an effective procedure for deciding whether
to accept new axioms.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 4:31:28 PM
Hero wrote:
> John Savard wrote:
> "Perhaps there may indeed be permanently unknowable truths."
> That's intriguing.
> Consider the following.
> The first  six digits of Pi
> 3.14159
> has five digits in the range 1 to 5
> (left hand)
> and one in the range 6 to 9 and 0
> (right hand).
> The following digit is a 2, that makes
> it  6 to 1 for left to right.
> Than a six, so right hand get's one more,
> makes it 6 to 2.
> One of the following statements is true:
> A) left wins
> B) right wins
> C) It will change, sometimes
> more digits in the lower range, and sometimes
> more in the higher.
> May be, we will never know the true
> answer

Maybe. It is an open question.

Maybe we will find the answer next year (don't bet on it), or maybe never. 
Event if there *is* a perfectly good answer, we might never find it. Or 
maybe there is no proof either way using methods we would consider sound.

> now i just need a proof of this.

Don't hold your breath.

Ralph Hartley
0
hartley (156)
1/11/2005 4:50:49 PM
In article <cs0u30038h@drn.newsguy.com>,
Daryl McCullough <daryl@atc-nycorp.com> wrote:
>That makes me think that the set of possible truths recognizable to
>mathematicians is indeed r.e. Mathematicians may make wild leaps in
>discovering their mathematical truth, but for their discoveries to be
>accepted, they have to be polishable to something that is recognizable
>as a mathematical proof.

The term under discussion was "permanently unknowable *truths*" (emphasis
mine).  The problem here is not just that theoremhood in certain axiomatic
systems is undecidable.  The problem is, at least in part, recognizing
which *axioms* are true.

The axiom of choice is now accepted as true by most mathematicians.
Did this acceptance hinge on polishing some kind of argument into
a proof?  Hardly.  As indicated by the term "axiom," the acceptance of
its truth is not based on proof (the trivial proof of an axiom from the
axiom itself is not the basis for anyone's believing in its truth).

If you object that not everyone accepts the axiom of choice as true, then
that just makes matters worse, because it suggests that the term "truths
that mathematicians accept" is too vague to admit precise mathematical
analysis.  Does V = L?  Is there a strongly inaccessible cardinal?
Some people say yes while others say no.  Does it even make sense to
predicate precise mathematical properties (such as "r.e.") of the set
of possible truths recognizable to mathematicians, if that "set" is so
ill-defined?
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 4:52:05 PM
daryl@atc-nycorp.com (Daryl McCullough) writes:

> That makes me think that the set of possible truths recognizable to
> mathematicians is indeed r.e. 

  Why do you think there is such a set?
0
torkel (478)
1/11/2005 4:55:46 PM
Torkel Franzen wrote:
> Ralph Hartley <hartley@aic.nrl.navy.mil> writes:
>> A is defined to include any procedure mathematicians *would* consider 
>> valid, including procedures that haven't been invented yet.
> 
> This is not a definition at all. What reason do you have to believe that
>  there is such an A?

I *did* say it was an assumption.

I certainly didn't claim to be able to prove it.

There are stronger versions of the CT thesis that imply it. For instance if 
the laws of physics are computable. Then any physical object, including
mathematicians, are effectively describable.

>> This is *not* a trivial assumption! It would be false, for example, if
>>  mathematicians had access to an oracle known to be infallible.
> 
> Oracles are irrelevant.

That was an *example* of how it could be false. It could be false in other 
ways.

> Mathematics changes and evolves, the opinion of mathematicians about 
> what is or is not a proof changes and evolves (in different directions).

But I am not convinced that is one of them.

The claim was not that there are facts no mathematician could ever have an 
opinion about, but that there are facts no mathematician could ever
*know*.

The word "know" implies some sort of objectivity. The claim that we can
know *anything* absolutely is a little white lie, but if you accept it at
all, then "know" must mean more than "in the opinion of mathematicians."

> Why do you think there is any such thing as "the procedures 
> mathematicians would consider valid"?

It is part of the fiction that mathematical truth is objective.

There *could* be objective methods that are not effective, but I don't know
of any. It is a reasonable, though by no means self evident, assumption 
that there are none.

If you don't consider the word "know" to imply "objectively", I won't argue 
semantics.

When you move away from abstract "mathematicians" and start talking about
real human beings, people are actually quite limited. I suspect there are
facts that *are* provable in ZFC, but no human can ever understand the proof.

Ralph Hartley
0
hartley (156)
1/11/2005 6:09:55 PM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> I *did* say it was an assumption.

  You said that "A is defined to include...". So what I'm questioning
is not an assumption - something that may or may not be true - but a
proposed definition. The question is whether you are in fact referring
to anything. I cannot sensibly say "Fnorx" and add that I'm assuming
that what I'm saying makes sense.

> There are stronger versions of the CT thesis that imply it. For instance if 
> the laws of physics are computable. Then any physical object, including
> mathematicians, are effectively describable.

  This makes no apparent sense. Let us suppose that people are
"effectively describable". This does not imply that it makes any sense
to speak of, for example, the astronomical or historical or
religious truths that people are capable of establishing or
perceiving. What is "established" or "perceived" is necessarily to a
large extent a matter of opinion.

> > Why do you think there is any such thing as "the procedures 
> > mathematicians would consider valid"?
> 
> It is part of the fiction that mathematical truth is objective.

  Not at all. To hold that mathematical truth, or historical truth,
or astronomical truth, is objective is not to hold that it is in the
least well-defined what today constitutes knowledge of mathematical,
historical, or astronomical truth, let alone that we can speak
sensibly of what people can know in principle in these fields.
People may hold all sorts of things to be true or evident or valid,
and opinions about what is true or evident or valid may vary due to
a number of subjective factors, even though the facts are
perfectly objective.
0
torkel (478)
1/11/2005 6:24:52 PM
tchow@lsa.umich.edu says...

>The axiom of choice is now accepted as true by most mathematicians.
>Did this acceptance hinge on polishing some kind of argument into
>a proof?  Hardly.

I think so.

>As indicated by the term "axiom," the acceptance of
>its truth is not based on proof (the trivial proof of an axiom from the
>axiom itself is not the basis for anyone's believing in its truth).

I didn't say proof, I said argument. An "argument" doesn't mean the same
thing as a "proof". You can give informal arguments for why you believe
in the axiom of choice, even if you can't prove it.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/11/2005 7:04:19 PM
In article <cs17vj011gu@drn.newsguy.com>, Daryl McCullough says...
>
>tchow@lsa.umich.edu says...
>
>>The axiom of choice is now accepted as true by most mathematicians.
>>Did this acceptance hinge on polishing some kind of argument into
>>a proof?  Hardly.
>
>I think so.

I'm sorry. I misspoke. What I meant was that I think that there exist arguments
for why the axiom of choice should be true. So the result of polishing is
something like a formal proof, together with informal arguments as to why the
axioms themselves are acceptable.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/11/2005 7:09:38 PM
In article <cs14gu$jk4$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>There are stronger versions of the CT thesis that imply it. For instance if 
>the laws of physics are computable. Then any physical object, including
>mathematicians, are effectively describable.

This is such a commonly stated fallacy that I think it's worth refuting it
in some detail.

By "the laws of physics are computable," I'll take you to mean something
roughly like the following: The universe is finite and discrete, with
one time dimension and some number of spatial dimensions; it has an
initial state, and for all t, the state at time t is determined by a
deterministic and recursive function of the states at times less than t.

The first objection is that many of these assumptions are not supported
experimentally (for example, that the universe is discrete, that it is
deterministic, that it has a well-defined initial state).  I'll ignore
this objection.

Observe that even under this scenario, the behavior of a particular human
being might depend on the whole universe.  Perhaps looking up at the stars
triggers some inspiration in my brain that would not have happened otherwise.
So we cannot assume that we can simplify the situation by restricting to,
say, the earth.  (I point this out not because it makes that much difference
to the argument, but because it is an obvious fact that, in spite of its
obviousness, seems to get overlooked by people who glibly make the
transition from the universe as a whole to people in particular.  This is
just to sound a warning note that the situation is not as trivial as it
might seem at first glance.)

The real point is that it does not follow from this model of the universe
that a given subset of the universe is going to be recursive, or even
recursively enumerable.  Let's take the mathematical truth "1+1=2."  What
does this truth correspond to in our model of the universe?  This is highly
unclear.  Do we search through the spacetime universe for things that
exhibit the geometric shape that "1+1=2" in Arabic numerals has?  Or do
we identify it with particular brain states of human beings?  Which brain
states and which parts of the brains correspond to "1+1=2"?  Are brain
states enough?  Maybe we need to capture the entire state of human society
for "1+1=2" to be classified as "human knowledge."

And even supposing that we could solve the knotty problem of identifying
mathematical truths with subsets of spacetime states, this does not
guarantee that such *subsets* are recursively enumerable, even if the
universe is.  Nor can we just take the universe as a whole and say
"It's somewhere in there," because if mathematical truths are somewhere
in the universe, then presumably mathematical falsehoods are as well,
and we have to be able to separate out the truths (in fact, the knowable
truths) from the falsehoods.

>There *could* be objective methods that are not effective, but I don't know
>of any.

Even theoremhood in ZFC is not effectively decidable, but it's surely
objective.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 7:21:39 PM
In article <cs189i012se@drn.newsguy.com>,
Daryl McCullough <daryl@atc-nycorp.com> wrote:
>I'm sorry. I misspoke. What I meant was that I think that there exist
>arguments for why the axiom of choice should be true. So the result
>of polishing is something like a formal proof, together with informal
>arguments as to why the axioms themselves are acceptable.

But your original argument was that "the ability to recognize solid
mathematics is as good as the ability to create it: You just enumerate
all possible character strings, and for each one, you check to see if
it is a solid mathematical argument."  What it seems that you've conceded
is that to recognize solid mathematics, you must also check to see if the
informal arguments for the axioms are correct.  We have no reason to
suppose that this can be done in any systematic manner even by professional
mathematicians, let alone students or AI programs.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 7:36:33 PM
tchow@lsa.umich.edu wrote:
> In article <cs14gu$jk4$1@ra.nrl.navy.mil>,
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> 
>>There are stronger versions of the CT thesis that imply it. For instance if 
>>the laws of physics are computable. Then any physical object, including
>>mathematicians, are effectively describable.
> 
> By "the laws of physics are computable," I'll take you to mean something
> roughly like the following: The universe is finite and discrete, with
> one time dimension and some number of spatial dimensions; it has an
> initial state, and for all t, the state at time t is determined by a
> deterministic and recursive function of the states at times less than t.

No, I don't mean roughly that.

Most, or perhaps all, of the assumptions you list are not necessary.

> The real point is that it does not follow from this model of the universe
> that a given subset of the universe is going to be recursive, or even
> recursively enumerable.

By your characterization of the thesis the universe only has finitely many 
states. You may have a definition of "subset of the universe" that allows 
it not to be finite, but that would be odd. All finite sets are recursive.

Not accepting that characterization, I don't think it matters, but it 
certianly shows that I have no idea what you could possibly mean.

> What does this truth correspond to in our model of the universe?  This is highly
> unclear.

The question is not what truth corresponds to. Perhaps you meant to say 
what does knowledge correspond to?

There are enough ways to parse definitions that I don't think that is an 
argument I want to have.

In any case it is mostly irrelevant to the main point:

If the laws of physics do not permit any physical process *whatsoever* to 
decide some question, then it is hard to see *any* sense in which 
inhabitants of that universe can be said to know the answer.

Of course the universe may have laws that don't have that property, or no 
laws at all, but the property is much less restrictive than being recursive.

>>There *could* be objective methods that are not effective, but I don't know
>>of any.
> 
> Even theoremhood in ZFC is not effectively decidable, but it's surely
> objective.

I said objective *methods*.

A proof in ZFC is an effective method.

Being a non-theorem in ZFC may be an objective fact, but it is hardly a 
method. It is one of those facts we might not be able to know.

Ralph Hartley
0
hartley (156)
1/11/2005 9:31:54 PM
Torkel Franzen wrote:
> Ralph Hartley <hartley@aic.nrl.navy.mil> writes:
> 
>>I *did* say it was an assumption.
> 
>   You said that "A is defined to include...". So what I'm questioning
> is not an assumption - something that may or may not be true - but a
> proposed definition. The question is whether you are in fact referring
> to anything. 

You knew *perfectly* well what I meant.

Are you questioning the assumption that something meeting the definition of 
A exists, or are you not?

> I cannot sensibly say "Fnorx" and add that I'm assuming
> that what I'm saying makes sense.

Are you claiming that "the effective procedure used by mathematicians to 
determine mathematical truth" is so nonsensical that it cannot *possibly* 
exist?

> To hold that mathematical truth, or historical truth,
> or astronomical truth, is objective is not to hold that it is in the
> least well-defined what today constitutes knowledge of mathematical,
> historical, or astronomical truth, let alone that we can speak
> sensibly of what people can know in principle in these fields.

Is it your position that the question "are there true facts that to which 
we can never know the answer?" is itself so ill formed, or poorly defined 
that it can not have a meaningful answer?

If so, why do you waste time on it?

> People may hold all sorts of things to be true or evident or valid,
> and opinions about what is true or evident or valid may vary due to
> a number of subjective factors, even though the facts are
> perfectly objective.

Here you verge into non-sequetir. I don't think I said anything about what 
"people may hold".

Given a set of 100 facts that I cannot determine the truth of, I can *hold* 
the correct evaluation of about 50 of them by guessing at random. That does 
not mean I *know* the truth of any of them.

Ralph Hartley
0
hartley (156)
1/11/2005 9:59:24 PM
tchow@lsa.umich.edu wrote:
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> 
>>I am only assuming mathematicians have *some* objective (i.e. effective) 
>>procedure for deciding what methods to accept.
>>
>>This is *not* a trivial assumption!
> 
> Indeed, more is true: on the face of it, the assumption seems obviously
> false.  Take, for example, "There exists a strongly inaccessible cardinal."
> Some mathematicians accept this as true; others don't.

But in what sense can any of them be said to *know* the answer?

I'm not even sure there even *is* an objective answer.

Before non-euclidean geometry was well understood, some mathematicians 
accepted "there exists a unique line through a given point parallel to a 
given line," and some did not. Is there any sense in which one group was 
right and the other wrong?

In fact, there is *another* unstated assumption: that mathematics is 
*sound*. It is quite likely that mathematicians accept some false things, 
even some contradictory things.

Ralph Hartley
0
hartley (156)
1/11/2005 10:19:04 PM
In article <cs1gbm$jsu$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>By your characterization of the thesis the universe only has finitely many 
>states.

Sorry, when I said that the universe is finite, I just meant that the
set of states at any given time is finite.  I meant for time to extend
indefinitely far into the future.

I also know that the definition I gave isn't necessarily close to what
you had in mind, but I picked it because it had what I think are the
key features that might make your claim plausible, sidestepping other
objections to your claim.  For example, I inserted discreteness because
if you allow a continuous universe represented by real numbers, then you
have to figure out what "computable" means, i.e., you have to specify a
model of real computation, and then we could argue about whether to use a
Blum-Shub-Smale model or a Grzegorczyk model---all of which is extraneous.
I made the universe finite because if at any given time it has infinitely
many states, then again there's the problem of defining a recursive
function with an infinite set of states as an argument.  I picked
determinism so as to avoid complications arising from quantum theory.
I picked a fixed initial state to avoid issues with possibly having a
singularity at t=0 where a nonrecursive amount of information could be
"hidden."  Etc.  These assumptions were picked so as to be favorable to
your point of view.

>In any case it is mostly irrelevant to the main point:
>If the laws of physics do not permit any physical process *whatsoever* to 
>decide some question, then it is hard to see *any* sense in which 
>inhabitants of that universe can be said to know the answer.

This is a totally different point from the one you made before, which is
what I was addressing.  Essentially, you were saying that if the laws of
physics are computable, then so would anything inside the universe,
including any method of coming to know truths.  Your new point is that
if the laws of physics do *not* allow computable processes, then there
cannot be a reasonable process for coming to know truths.  These are
totally different assertions.

Getting back to your original assertion, my point is that even if the laws
of physics are computable, it doesn't follow that methods of acquiring
knowledge "inside" that universe are computable.  I agree that what I said
is irrelevant to your new "main point," but I wasn't addressing your new
"main point."

As for your new main point, see below.

>Being a non-theorem in ZFC may be an objective fact, but it is hardly a 
>method. It is one of those facts we might not be able to know.

I propose the following method of acquiring truth:

1. Generate conjectures in some systematic way (e.g., stated in the
first-order language of set theory).

2. Search for proofs or disproofs of the conjectures from the axioms of ZFC
by exhaustive enumeration.

3. Declare success when you find such a proof or disproof.

This procedure is not effective, because theoremhood is r.e. and not
recursive.  Does that make this method of acquiring truth not objective?
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 11:34:26 PM
In article <cs1j43$jup$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>tchow@lsa.umich.edu wrote:
>> Indeed, more is true: on the face of it, the assumption seems obviously
>> false.  Take, for example, "There exists a strongly inaccessible cardinal."
>> Some mathematicians accept this as true; others don't.
>
>But in what sense can any of them be said to *know* the answer?

Ah, I'm beginning to see better where you're coming from.  So let me back up
a bit.

Let's take something like Brouwer's fixed-point theorem.  Do we know that
Brouwer's fixed-point theorem is true?  Well, it's a standard theorem of
mathematics.  It is proved in textbooks.  The proof can be formalized in
ZFC.  It can be formalized in much weaker axiomatic systems, including
one known as WKL_0.  Mathematicians all accept it as true.  They would
say that they know it's true.  Are they right?

Well, when I said that mathematicians "all" accept is as true, I lied.
Brouwer himself came to reject it as using illegitimate "non-constructive"
methods.  Intuitionists reject classical logic, in particular the law of
the excluded middle, so they reject many classically valid proofs.

If disagreement among mathematicians is enough to render suspect anything
purporting to be "knowledge," then we have to jettison Brouwer's fixed-point
theorem.  But there is worse in store.  Some mathematicians are formalists
or finitists.  They reject even more of standard mathematics.  Some of them
reject the axiom of infinity of ZFC.  So I guess we don't know all the
axioms of ZFC either.

In fact, do we know *any* of the axioms of ZFC?  Or of PA?  Or of any other
axiomatic system you might write down?  Maybe you can get everyone to agree
on some axioms, but is agreement really enough to yield *knowledge*?  Surely
not.  We could all agree on something that is false, right?

We could take the extreme skeptical view, that we cannot know anything.
Then to deduce the existence of permanently unknowable truths, all we
need is the existence of *some* truth (which perforce is permanently
unknowable).  But I doubt that you are advocating such an extreme position.
Most likely you think we do know some things---"finitary" statements
perhaps, such as those formalizable in primitive recursive arithmetic
(PRA).  Do you think that PRA exhausts mathematical knowledge?  This
would be a defensible position at least, although you would be rejecting
most of what mathematicians claim to "know."  But getting back to the
question of "permanently unknowable truths," it's not clear to me how
you would be able to establish their existence.  PRA isn't strong enough
to prove Goedel's theorems, for example.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/11/2005 11:54:43 PM
tchow@lsa.umich.edu wrote:
> In article <cs0u30038h@drn.newsguy.com>,
> Daryl McCullough <daryl@atc-nycorp.com> wrote:
> >That makes me think that the set of possible truths recognizable to
> >mathematicians is indeed r.e. Mathematicians may make wild leaps in
> >discovering their mathematical truth, but for their discoveries to
be
> >accepted, they have to be polishable to something that is
recognizable
> >as a mathematical proof.
>
> The term under discussion was "permanently unknowable *truths*"
(emphasis
> mine).  The problem here is not just that theoremhood in certain
axiomatic
> systems is undecidable.  The problem is, at least in part,
recognizing
> which *axioms* are true.
>

This seems silly to me.  Suppose someone asks, "Are the group axioms
true?"  Obviously, they are true of any set of objects which satisfy
them.  But suppose he asks again, emphatically, "No, but are they
*true*?"  What is that to mean?

> The axiom of choice is now accepted as true by most mathematicians.

I don't think "as true" is necessary here.  In fact, I think it
obfuscates the issue.  The acceptance of AC is as much a consequence of
sociology as it is of logical consistency.

'cid 'ooh

0
poopdeville (133)
1/12/2005 1:02:30 AM
In article <41e46743$0$575$b45e6eb0@senator-bedfellow.mit.edu>,
 <tchow@lsa.umich.edu> wrote:
>PRA isn't strong enough to prove Goedel's theorems, for example.

According to Shankar's _ Mathematics, Machines, and Goedel's Proof _;

  ``Goedel's incompleteness theorem represents a significant landmark in
  mathematics. [...] Feferman [Fe82] has shown that the incompleteness
  proof can be carried out within PRA.'' (page 141)

[Fe82] S. Feferman. Inductively presented systems and the formalization of
mathematics. In D. van Dalen, D. Lascar, and J. Smiley, editors, _ Logic
Colloquium '80 _. North-Holland, Amsterdam, 1982.
-- 
Josh Purinton
0
1/12/2005 1:33:34 AM
---------------------------------------------s-o-s-----------------------------------------
"Ralph Hartley" <hartley@aic.nrl.navy.mil> wrote in
> Hero wrote:
> > John Savard wrote:
> > "Perhaps there may indeed be permanently unknowable truths."
> > That's intriguing.
> > Consider the following.
> > The first  six digits of Pi
> > 3.14159
> > has five digits in the range 1 to 5
> > (left hand)
> > and one in the range 6 to 9 and 0
> > (right hand).
> > The following digit is a 2, that makes
> > it  6 to 1 for left to right.
> > Than a six, so right hand get's one more,
> > makes it 6 to 2.
> > One of the following statements is true:
> > A) left wins
> > B) right wins
> > C) It will change, sometimes
> > more digits in the lower range, and sometimes
> > more in the higher.
> > May be, we will never know the true
> > answer
>
> Maybe. It is an open question.
>
> Maybe we will find the answer next year (don't bet on it), or maybe never.
> Event if there *is* a perfectly good answer, we might never find it. Or
> maybe there is no proof either way using methods we would consider sound.
>
> > now i just need a proof of this.
>
> Don't hold your breath.

Exactly, they cross over with P=1 but expected time is oo.
See http://www.ms.uky.edu/~mai/java/stat/brmo.html  on random walks.

Herc



0
erc
1/12/2005 1:37:52 AM
Tim Chow:

>PRA isn't strong enough
>to prove Goedel's theorems, for example.

Happy new year.
PRA is strong enough, which is what lies behind the Goedel-von Neumann
second incompleteness theorem (i.e., the first incompleteness theorem can be
formalized in a weak subsystem of arithmetic).

Easier to think in terms of ISigma_1 which conservatively extends PRA.
ISigma_1 is the subsystem of PA with induction restricted to Sigma_1
formulas phi(x, y1, ..., yn) (allowing parameters y1, ..., yn).
ISigma_1 is enough because you only need to prove things about provability,
and provability is r.e. (and thus Sigma_1 definable).
That ISigma_1 is a conservative extension of PRA is due to Parsons 1970 (who
says philosophers never prove interesting mathematical results!), but I've
never worked through the details.

Suppose T is an axiomatizable theory extending Q and G is a Goedel-Rosser
sentence for T.
Then ISigma_1 proves the first incompleteness result. I.e., it proves "if T
is consistent then G is undecidable in T".

Sketch of details (for more, see Hajek/Pudlak 1991, _Metamathematics of
First-Order Arithmetic_):

1. Let T be an axiomatizable theory in some countable first-order L, which
is goedel-coded into N. For expression E, let #E be its code, and [E] be its
canonical numeral.
2. Let Proof_T(x, y) be some definition of "x is the code of a proof of a
formula whose code is y in T" satisfying Hilbert-Bernays derivability
conditions (to block funny provability predicates).
3. Let Prov_T(x) be Ey Proof_T(y, x).
4. Let G be a Goedel-Rosser fixed-point sentence relative to this
provability predicate.
5. Let Con(T) be "there is no proof in T of 0=1".

Then, the 1st incompleteness theorem says "If T is consistent, then G is
undecidable in T". This can be proved in ISigma_1. I.e.,

   (*) ISigma_1 |- Con(T) -> (~Prov_T([G]) & ~Prov_T([~G]))

Thus,

   (**) ISigma_1 |- Con(T) -> G

Second Incompleteness Theorem: If T is consistent, then T does not prove
Con(T).
Proof: If T extends ISigma_1 and T proves Con(T), then (by (**)), T proves
G. But, by the First incompleteness theorem, if T is consistent, then T does
not prove G. Hence, if T is consistent, then T does not prove Con(T).

This finitary reasoning can itself be formalized within ISigma_1. Thus,

  ISigma_1 |- Con(T) -> ~Prov_T([Con(T])

(i.e., we have a finitary proof of "if T is consistent, then Con(T) is not
provable in T.")

--- Jeff




0
ketland (18)
1/12/2005 2:11:35 AM

tchow@lsa.umich.edu wrote:

>In article <2IIEd.47383$8l.9575@pd7tw1no>,
>namducnguyen  <namducnguyen@shaw.ca> wrote:
>  
>
>>Imho, if after 70+ years since Godel's work and "there isn't one" such
>>definition, then hinting that Godel's work "indirectly" has something to
>>do with "permanently unknownable truths" would be quite ...revealing -
>>though it might be controversial. And you're going to keep it all silent,
>>lest that something silly "might" be said?
>>    
>>
>
>I don't understand your question.  Are you suggesting that I, personally,
>have something concrete I want to say about "permanently unknowable truths"
>and am just afraid to say it?  That's not true, and I never said as much.
>  
>

On my first question:

 >>1) As far as FOL is concerned, what is the definition of "_permanently_
 >>unknownable truth"?
 >

you gave a very "strong" answer:

 >
 >There isn't one.

On my 2nd question:

 >>2) What does Godel's work has to do with "permanently unknownable 
truths"?
 >

the answer "Nothing directly" sorts of implies that  _indirectly_ there 
might be some
relationship between Godel's work and the "permanently unknownable 
truths" - at least
that's what it seemed to sound to me. Any rate, it's just  probably a 
mis-interpretation
on my part of what you intended to say. Sorry about that.

 >
 >Nothing directly.  Of course in any respectable attempt to discuss 
what that
 >term could possibly mean, a good understanding of Goedel's work would be
 >useful, if only to avoid saying something obviously silly.
0
1/12/2005 6:53:36 AM
> On my first question:
>
>  >>1) As far as FOL is concerned, what is the definition of "_permanently_
>  >>unknownable truth"?
>  >
>
> you gave a very "strong" answer:
>
>  >
>  >There isn't one.
>
> On my 2nd question:
>
>  >>2) What does Godel's work has to do with "permanently unknownable
> truths"?
>  >
>
> the answer "Nothing directly" sorts of implies that  _indirectly_ there
> might be some
> relationship between Godel's work and the "permanently unknownable
> truths" - at least
> that's what it seemed to sound to me. Any rate, it's just  probably a
> mis-interpretation
> on my part of what you intended to say. Sorry about that.



I would suggest that something can exist and not be defined *yet*, hence
there is only a weak relationship to 'well' defined theory.  In which case Chow's
2 answers are mutually supportive.

Herc



0
erc
1/12/2005 7:19:58 AM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> Given a set of 100 facts that I cannot determine the truth of, I can *hold* 
> the correct evaluation of about 50 of them by guessing at random. That does 
> not mean I *know* the truth of any of them.

  Precisely! So what distinction between knowing and mere holding do
you have in mind with reference to mathematical statements? Indeed we
can be more specific: with reference to arithmetical statements of the
form "for every n, P(n)" where P is a recursive predicate?


0
torkel (478)
1/12/2005 7:52:51 AM
poopdeville@gmail.com writes:

> But suppose he asks again, emphatically, "No, but are they
> *true*?"  What is that to mean?

  But nobody has asked that.

> I don't think "as true" is necessary here.  In fact, I think it
> obfuscates the issue.  The acceptance of AC is as much a consequence of
> sociology as it is of logical consistency.

  Trivially, logical consistency is insufficient for a mathematical
axiom to be accepted.


0
torkel (478)
1/12/2005 7:54:29 AM
"Torkel Franzen" <torkel@sm.luth.se> ha scritto

> > That makes me think that the set of possible truths recognizable to
> > mathematicians is indeed r.e.
>
>   Why do you think there is such a set?

There is not a recursive procedure to assign names to costructive Ordinals.
We could obtain a NON-r.e. set of "possible recognizable truths" by
collecting the sentances "A is an ordinal" for any costructive ordinal... am
I right?


0
1/12/2005 11:36:28 AM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:


> There is not a recursive procedure to assign names to costructive Ordinals.
> We could obtain a NON-r.e. set of "possible recognizable truths" by
> collecting the sentances "A is an ordinal" for any costructive
> ordinal...

  What I was wondering was on what grounds Daryl took the notion of
"possible truth recognizable to mathematicians" to be well-defined.
What does it mean to "recognize" a mathematical truth, say of the
form "for every natural number n, P(n)", where P is a mechanically
decidable property.


0
torkel (478)
1/12/2005 11:39:17 AM
Torkel Franzen says...
>
>"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:
>
>
>> There is not a recursive procedure to assign names to costructive Ordinals.
>> We could obtain a NON-r.e. set of "possible recognizable truths" by
>> collecting the sentances "A is an ordinal" for any costructive
>> ordinal...
>
>  What I was wondering was on what grounds Daryl took the notion of
>"possible truth recognizable to mathematicians" to be well-defined.
>What does it mean to "recognize" a mathematical truth, say of the
>form "for every natural number n, P(n)", where P is a mechanically
>decidable property.

I think you're making it more mysterious than it actually is. There
really are only a handful of tricks that mathematicians use to
try to determine the truth of a mathematical statement, and they
seem to be pretty much captured by ZFC, together with reflection-type
principles ("I believe ZFC, so Con(ZFC) is not much of a stretch".

Is there any reason to think that ZFC (or reflection-inspired
extensions) *doesn't* cover what mathematicians believe to be
solid mathematics?

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 11:53:01 AM
daryl@atc-nycorp.com (Daryl McCullough) writes:

> I think you're making it more mysterious than it actually is. There
> really are only a handful of tricks that mathematicians use to
> try to determine the truth of a mathematical statement, and they
> seem to be pretty much captured by ZFC, together with reflection-type
> principles ("I believe ZFC, so Con(ZFC) is not much of a stretch".

  So by "the set of possible truths recognizable to mathematicians"
you mean "the set of truths provable in ZFC or in a reflection-type
extension of ZFC". Two questions naturally arise. First, why should
we adopt this stipulation of what "recognizable truth" means? Second,
even given this stipulation, you haven't actually defined any set
for which it makes any sense to ask whether or not it is recursively
enumerable. "ZFC plus reflection-type principles" is too vague for
this.
0
torkel (478)
1/12/2005 12:18:39 PM
"Torkel Franzen" <torkel@sm.luth.se> ha scritto

> > There is not a recursive procedure to assign names to costructive
Ordinals.
> > We could obtain a NON-r.e. set of "possible recognizable truths" by
> > collecting the sentances "A is an ordinal" for any costructive
> > ordinal...
>
>   What I was wondering was on what grounds Daryl took the notion of
> "possible truth recognizable to mathematicians" to be well-defined.

Ok, and I was suggesting a way to prove that there is a set of "possible
truths recognizable" that is NOT r.e.


0
1/12/2005 1:13:23 PM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:


> Ok, and I was suggesting a way to prove that there is a set of "possible
> truths recognizable" that is NOT r.e.

  What do you intend by describing the statements of the form you
indicated as "possible recognizable truths"?




0
torkel (478)
1/12/2005 1:19:41 PM
tchow@lsa.umich.edu wrote:
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> 
>>There are stronger versions of the CT thesis that imply it. For instance if 
>>the laws of physics are computable. Then any physical object, including
>>mathematicians, are effectively describable.
> 
> By "the laws of physics are computable," I'll take you to mean something
> roughly like the following: The universe is finite and discrete, with
> one time dimension and some number of spatial dimensions; it has an
> initial state, and for all t, the state at time t is determined by a
> deterministic and recursive function of the states at times less than t.

That wasn't close. Perhaps, I should have been more clear about what I 
*did* mean.

Physical CT thesis (my definition): The laws of physics do not admit any 
computing device more powerful than a TM.

Or: No physical system can compute any non-recursive function.

There should be some fine print about what it means for a physical system 
to "compute" a function, but the details of a good definition shouldn't matter.

Assume that, given an input, you can construct an initial state by any 
recursive process, let the physical system evolve, and use any recursive 
process (with access to the entire history if needed) to decode the result 
(This could be stated without reference to time evolution, but it is 
trickier). States of the system may be represented in any reasonable way.

The thesis is not a tautology. Many theories include physical constants 
that must be measured, not calculated. Those real numbers could be 
recursive, or not. It is quite likely that they are essentially random 
(which technically makes them non-recursive, but not good for much). In 
principle, they could even permit devices that solve the halting problem, 
e.g. if the fine structure constant encoded Chaitin's omega.

With that caveat, I don't know of any real physical theory that permits the 
computation of non-recursive functions.

Ralph Hartley
0
hartley (156)
1/12/2005 1:22:07 PM
tchow@lsa.umich.edu wrote:

> I also know that the definition I gave isn't necessarily close to what
> you had in mind, but I picked it because it had what I think are the
> key features that might make your claim plausible, sidestepping other
> objections to your claim.  For example, I inserted discreteness because
> if you allow a continuous universe represented by real numbers, then you
> have to figure out what "computable" means, i.e., you have to specify a
> model of real computation, and then we could argue about whether to use a
> Blum-Shub-Smale model or a Grzegorczyk model---all of which is extraneous.
> I made the universe finite because if at any given time it has infinitely
> many states, then again there's the problem of defining a recursive
> function with an infinite set of states as an argument.  I picked
> determinism so as to avoid complications arising from quantum theory.
> I picked a fixed initial state to avoid issues with possibly having a
> singularity at t=0 where a nonrecursive amount of information could be
> "hidden."  Etc.  These assumptions were picked so as to be favorable to
> your point of view.

I might have accepted this explanation if you hadn't said:

> The first objection is that many of these assumptions are not supported
> experimentally

>Ralph Hartley said:
>>In any case it is mostly irrelevant to the main point:
>>If the laws of physics do not permit any physical process *whatsoever* to 
>>decide some question, then it is hard to see *any* sense in which 
>>inhabitants of that universe can be said to know the answer.
> 
> This is a totally different point from the one you made before, which is
> what I was addressing.

But it is exactly the point I was *trying* to make before. Clearly, I 
failed, since it is totally different from the point you thought I was 
making. I still don't understand exactly what you thought I was claiming, 
and what your objection to it was, but it's a moot point.

In retrospect, perhaps I should not have brought it up at all, unless I was 
willing to spend more time on it.

> Your new point is that
> if the laws of physics do *not* allow computable processes, then there
> cannot be a reasonable process for coming to know truths.  These are
> totally different assertions.

That still isn't it.

My point is that if the laws of physics do not allow *non* computable 
processes ...

Perhaps, that was a typo on your part.

>>Being a non-theorem in ZFC may be an objective fact, but it is hardly a 
>>method. It is one of those facts we might not be able to know.
> 
> I propose the following method of acquiring truth:
> 
> 1. Generate conjectures in some systematic way (e.g., stated in the
> first-order language of set theory).
> 
> 2. Search for proofs or disproofs of the conjectures from the axioms of ZFC
> by exhaustive enumeration.
> 
> 3. Declare success when you find such a proof or disproof.
> 
> This procedure is not effective, because theoremhood is r.e. and not
> recursive.  Does that make this method of acquiring truth not objective?

No. Perhaps "effective" is not exactly the word I intended to use? I 
intended to include *partial* recursive functions. Does "effective" have 
such a precice definition that it excludes them?

Ralph Hartley
0
hartley (156)
1/12/2005 2:13:01 PM
|-|erc wrote:

> Exactly, they cross over with P=1 but expected time is oo.
> See http://www.ms.uky.edu/~mai/java/stat/brmo.html  on random walks.

That would be true if the digits of PI were random.

As it is, it is an open question if all the digits even occur infinitely 
many times.

There *could* be (but I doubt it) an N such that the digits 1-5 *never* 
appear after position N. In that case right would surely win.

Or the digits could *alternate* between left hand and right hand (for 
positions greater than some N).

Really, very little is known for sure about most of the digits of pi (in 
any base).

It isn't at all clear what you even *mean* by "P=1" when talking about 
digits of pi, since probability theory doesn't directly apply. If you mean 
"what odds I would consider fair in a bet", I would say P~=1. There is a 
remote possibility that the digits of pi might do something "funny", but it 
could go either way.

Before taking the bet I would also ask how the answer is to be settled. I 
would *not* allow a third party to hold the stakes indefinitely or without 
interest :-).

Ralph Hartley
0
hartley (156)
1/12/2005 2:46:15 PM
Torkel Franzen says...

>  So by "the set of possible truths recognizable to mathematicians"
>you mean "the set of truths provable in ZFC or in a reflection-type
>extension of ZFC". Two questions naturally arise. First, why should
>we adopt this stipulation of what "recognizable truth" means?

I wasn't stipulating. It was an empirical claim. All the evidence
seems to support the claim that anything that mathematicians accept
as true is captured by ZFC or some extensions.

>Second, even given this stipulation, you haven't actually defined
>any set for which it makes any sense to ask whether or not it is
>recursively enumerable. "ZFC plus reflection-type principles" is
>too vague for this.

Then forget reflection. I'm making the empirical conjecture
that for mathematical problems arising in fields outside of
mathematical logic and set theory, if they are humanly solvable
at all, then the solution can be formulated as a proof in ZFC.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 3:12:37 PM
daryl@atc-nycorp.com (Daryl McCullough) writes:

> I wasn't stipulating. It was an empirical claim. All the evidence
> seems to support the claim that anything that mathematicians accept
> as true is captured by ZFC or some extensions.

  You mean anything that is at the moment accepted as true by most
mathematicians? ZFC does indeed play such a role, although many
mathematicians would reject the claim that every arithmetical
statement provable in ZFC can be recognized as true. But from this
empirical observation we can draw no large conclusions about what
is a "recognizable truth". For example, are the consequences of
ZFC+"There is an inaccessible cardinal" recognizable truths?

> Then forget reflection. I'm making the empirical conjecture
> that for mathematical problems arising in fields outside of
> mathematical logic and set theory, if they are humanly solvable
> at all, then the solution can be formulated as a proof in ZFC.

  Again this can not be described as a conjecture at all as long as
you haven't given content to "humanly solvable". What does this mean?
Are problems that can be solved using axioms of infinity (as above)
humanly solvable?
0
torkel (478)
1/12/2005 3:36:39 PM
Torkel Franzen says...

>> Then forget reflection. I'm making the empirical conjecture
>> that for mathematical problems arising in fields outside of
>> mathematical logic and set theory, if they are humanly solvable
>> at all, then the solution can be formulated as a proof in ZFC.
>
>  Again this can not be described as a conjecture at all as long as
>you haven't given content to "humanly solvable". What does this mean?

I don't understand what you think is unclear. The meaning of an
empirical claim is its falsification conditions. To falsify my
claim, it is enough that (1) at some future date, some problem arising
from physics, number theory, topology, or any scientific or mathematical
field other than set theory and mathematical logic, (2) at a yet later
date, mathematicians come to the consensus about the solution to the
problem, and (3) at yet a later date, someone proves that the solution
cannot be formalized within ZFC.

>Are problems that can be solved using axioms of infinity (as above)
>humanly solvable?

Sure.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 4:10:37 PM
In article <cs2191$1pf$1@newsg2.svr.pol.co.uk>,
Jeffrey Ketland <ketland@ketland.fsnet.co.uk> wrote:
>Tim Chow:
>>PRA isn't strong enough to prove Goedel's theorems, for example.
>
>PRA is strong enough

Yes, sorry.  I realized my gaffe just after posting.  Seems my anti-spam
mechanisms also prevent me from canceling my own messages.

More generally, (a version of) Goedel's incompleteness theorem is provable
in RCA_0, and Goedel's completeness theorem is provable in WKL_0, and WKL_0
is conservative over PRA for Pi^0_2 sentences.  I think Goedel's theorems
can be formulated as Pi^0_2 sentences.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 5:14:40 PM
daryl@atc-nycorp.com (Daryl McCullough) writes:

>>Are problems that can be solved using axioms of infinity (as above)
>>humanly solvable?
> 
> Sure.

  Why is that? Just what does "humanly solvable" mean?

  
0
torkel (478)
1/12/2005 5:16:54 PM
In article <1105491750.589150.193360@c13g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>This seems silly to me.  Suppose someone asks, "Are the group axioms
>true?"  Obviously, they are true of any set of objects which satisfy
>them.  But suppose he asks again, emphatically, "No, but are they
>*true*?"  What is that to mean?

First of all, if truth doesn't mean anything, then there are certainly no
truths at all, let alone permanently unknowable ones.

But more to the point, group axioms aren't the right example to choose
here.  I might ask, is it true that every differentiable function is
continuous?  Is it true that the square root of 2 is irrational?  We
all say "yes" in ordinary mathematical practice.  That's because we
can prove them.  But proof is only useful in establishing truth if the
axioms that it starts with are true.  Whatever axioms you choose to
start with (ZFC, 2nd order arithmetic, whatever), the question arises
as to whether those axioms are true.  If you say that asking about the
truth of the axioms is meaningless, then presumably asking about the
truth of the theorems proved from them is also meaningless, and we're
back in the situation where there are no truths at all.

>> The axiom of choice is now accepted as true by most mathematicians.
>
>I don't think "as true" is necessary here.  In fact, I think it
>obfuscates the issue.  The acceptance of AC is as much a consequence of
>sociology as it is of logical consistency.

But in the context of the question of the existence of permanently
unknowable truths, is AC a known truth?  Is it true that every vector
space has a basis?  If these things aren't "true," then what is?
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 5:26:35 PM
tchow@lsa.umich.edu wrote:
> In article <1105491750.589150.193360@c13g2000cwb.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> 
>>This seems silly to me.  Suppose someone asks, "Are the group axioms
>>true?"  Obviously, they are true of any set of objects which satisfy
>>them.  But suppose he asks again, emphatically, "No, but are they
>>*true*?"  What is that to mean?
> 
> 
> First of all, if truth doesn't mean anything, then there are certainly no
> truths at all, let alone permanently unknowable ones.
> 
> But more to the point, group axioms aren't the right example to choose
> here.  I might ask, is it true that every differentiable function is
> continuous?  Is it true that the square root of 2 is irrational?  We
> all say "yes" in ordinary mathematical practice.  That's because we
> can prove them.  But proof is only useful in establishing truth if the
> axioms that it starts with are true.  Whatever axioms you choose to
> start with (ZFC, 2nd order arithmetic, whatever), the question arises
> as to whether those axioms are true.  If you say that asking about the
> truth of the axioms is meaningless, then presumably asking about the
> truth of the theorems proved from them is also meaningless, and we're
> back in the situation where there are no truths at all.
> 

Not at all. Instead of worrying about "are they true?" one starts with 
the axioms and sees what you can prove with them. In other words, ZFC is 
a model. Within that model one can prove certain things. The question is 
really whether it is useful or not. So far, while AC has led to some 
bizarre results (Banach-Tarksi, for one) it is also equivalent to some 
things that prove quite useful, e.g. "Every vector space has a basis", 
"The arbitrary product of compact spaces is compact". Thus many 
mathematicians are quite happy to include the Axiom of Choice in their 
model of set theory.
> 
>>>The axiom of choice is now accepted as true by most mathematicians.
>>
>>I don't think "as true" is necessary here.  In fact, I think it
>>obfuscates the issue.  The acceptance of AC is as much a consequence of
>>sociology as it is of logical consistency.
> 
> 
> But in the context of the question of the existence of permanently
> unknowable truths, is AC a known truth?  Is it true that every vector
> space has a basis?  If these things aren't "true," then what is?
0
1/12/2005 5:39:10 PM
Ron Sperber <ronsperber@optonline.net> writes:

> In other words, ZFC is  a model.

  In mathematical terminology, ZFC is not a model but a formal theory.
What does the above statement mean?
0
torkel (478)
1/12/2005 5:54:11 PM
I asked:

> >>Are problems that can be solved using axioms of infinity (as above)
> >>humanly solvable?
> > 
> > Sure.
> 
>   Why is that? Just what does "humanly solvable" mean?

  To save time, I should perhaps explicitly point out that you earlier
said about mathematical problems that "if they are humanly solvable
at all, then the solution can be formulated as a proof in ZFC". Since
problems solvable using axioms of infinity are not as a rule solvable
in ZFC, it's difficult to know what to make of your statements.

0
torkel (478)
1/12/2005 6:21:53 PM
"Torkel Franzen" <torkel@sm.luth.se> ha scritto

> > Ok, and I was suggesting a way to prove that there is a set of "possible
> > truths recognizable" that is NOT r.e.
>
>   What do you intend by describing the statements of the form you
> indicated as "possible recognizable truths"?

It is not so relevant waht is intended exactly: it suffices we agree in
considering a statement "A is an ordinal" as a "possible recognizable truth"
whenever A is a costructible ordinal... is it problematc?


0
1/12/2005 6:47:56 PM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:

> It is not so relevant waht is intended exactly: it suffices we agree in
> considering a statement "A is an ordinal" as a "possible recognizable truth"
> whenever A is a costructible ordinal... is it problematc?

  I have no idea, since you have given no explanation of what
"possible recognizable truth" means. Are arithmetical statements in
general "possible recognizable truths"? If not, why not? What's special
about the statements you indicate?
0
torkel (478)
1/12/2005 6:51:27 PM
Torkel Franzen wrote:
> Ron Sperber <ronsperber@optonline.net> writes:
> 
> 
>>In other words, ZFC is  a model.
> 
> 
>   In mathematical terminology, ZFC is not a model but a formal theory.
> What does the above statement mean?
How is it not a model for set theory? In other words, ZFC describes 
axioms that sets obey. If we start with these axioms, one can prove 
somethings and disprove some things, and fail to be able to do either 
with some others.
0
1/12/2005 6:52:53 PM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:

> It is not so relevant waht is intended exactly: it suffices we agree in
> considering a statement "A is an ordinal" as a "possible recognizable truth"
> whenever A is a costructible ordinal... is it problematc?

  I have no idea, since you haven't explained what "possible
recognizable truth" means.


0
torkel (478)
1/12/2005 6:53:39 PM
Ron Sperber <ronsperber@optonline.net> writes:

> How is it not a model for set theory?

  In ordinary mathematical terminology, it makes no sense whatever to
say that ZFC is a model for set theory.

> In other words, ZFC describes 
> axioms that sets obey.

  So by saying that "ZFC is a model", do you mean that the axioms of
ZFC are true?
0
torkel (478)
1/12/2005 6:54:53 PM
<poopdeville@gmail.com> ha scritto

> > which *axioms* are true.
> >
>
> This seems silly to me.  Suppose someone asks, "Are the group axioms
> true?"  Obviously, they are true of any set of objects which satisfy
> them.  But suppose he asks again, emphatically, "No, but are they
> *true*?"  What is that to mean?

When we ask if a (formal) statement is true, we mean "true in a specific
model" that we may not specify if it is clear which the model is. If the
statements are the group axioms it is unclear what is the model, not if the
statement is the axiom of choice.


0
1/12/2005 7:04:16 PM
Torkel Franzen says...

>  To save time, I should perhaps explicitly point out that you earlier
>said about mathematical problems that "if they are humanly solvable
>at all, then the solution can be formulated as a proof in ZFC". Since
>problems solvable using axioms of infinity are not as a rule solvable
>in ZFC, it's difficult to know what to make of your statements.

I don't know what to make of yours, either. What do you mean by
axioms of infinity?

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 7:09:08 PM
"Ron Sperber" <ronsperber@optonline.net> ha scritto

> How is it not a model for set theory? In other words, ZFC describes
> axioms that sets obey. If we start with these axioms, one can prove
> somethings and disprove some things, and fail to be able to do either
> with some others.

ZFC is a set of axiom (and theorems) that may be true in different models
(for example the universe of costructible sets is one of them)


0
1/12/2005 7:09:22 PM
"Torkel Franzen" <torkel@sm.luth.se> ha scritto

> > It is not so relevant waht is intended exactly: it suffices we agree in
> > considering a statement "A is an ordinal" as a "possible recognizable
truth"
> > whenever A is a costructible ordinal... is it problematc?
>
>   I have no idea, since you haven't explained what "possible
> recognizable truth" means.

We are making some philosophy, not formal mathematics, so why so many
problem for the use of intuitive concepts?
Did Church and Turing explain what they ment by "intuitively computable"
when they formulated their thesis?


0
1/12/2005 7:14:48 PM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:

> Did Church and Turing explain what they ment by "intuitively computable"
> when they formulated their thesis?

  Not "intuitively computable", but "mechanically computable". And
certainly they did. Nothing similar has been presented for "possible
recognizable truth".
0
torkel (478)
1/12/2005 7:18:36 PM
In article <vcb6522ld8e.fsf@beta19.sm.ltu.se>, Torkel Franzen says...
>
>I asked:
>
>> >>Are problems that can be solved using axioms of infinity (as above)
>> >>humanly solvable?
>> > 
>> > Sure.

Sorry, I did not understood that you were using "axioms of infinity" to
mean "large cardinal axioms". I don't see how proving something is true
under the assumption that there exists a particular type of large cardinal
is any different from proving any other conditional statement.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 7:18:45 PM
On Wed, 12 Jan 2005 13:52:53 -0500, Ron Sperber wrote:
> Torkel Franzen wrote:
>> Ron Sperber <ronsperber@optonline.net> writes:
>> 
>> 
>>>In other words, ZFC is  a model.
>> 
>> 
>>   In mathematical terminology, ZFC is not a model but a formal theory.
>> What does the above statement mean?
> How is it not a model for set theory? In other words, ZFC describes 
> axioms that sets obey. If we start with these axioms, one can prove 
> somethings and disprove some things, and fail to be able to do either 
> with some others.

Do the axioms for a group form a model for group theory?  Which of the
axioms serves as the identity in this model?


-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
0
dseaman (1174)
1/12/2005 8:16:38 PM
daryl@atc-nycorp.com (Daryl McCullough) writes:

> I don't see how proving something is true
> under the assumption that there exists a particular type of large cardinal
> is any different from proving any other conditional statement.

Conditional statements? What conditional statements?  Your statement
was that any problem that can be solved using axioms of infinity
is humanly solvable. Why is this?
0
torkel (478)
1/12/2005 8:21:07 PM
Torkel Franzen says...
>
>daryl@atc-nycorp.com (Daryl McCullough) writes:
>
>> I don't see how proving something is true
>> under the assumption that there exists a particular type of large cardinal
>> is any different from proving any other conditional statement.
>
>Conditional statements? What conditional statements? Your statement
>was that any problem that can be solved using axioms of infinity
>is humanly solvable.

I didn't say that. I said "Sure". But that was under a mistaken impression
of what you meant by "axioms of infinity".

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 8:32:35 PM
"Torkel Franzen" <torkel@sm.luth.se> ha scritto

> > Did Church and Turing explain what they ment by "intuitively computable"
> > when they formulated their thesis?
>
>   Not "intuitively computable", but "mechanically computable". And
> certainly they did. Nothing similar has been presented for "possible
> recognizable truth".

Turing words:

"Every 'function which would naturally be regarded as computable' can be
computed by a Turing machine"

The expression 'function which would naturally be regarded as computable' is
intentionally left vague and not formally defined.

In the same way I wanted to say something about some "statements that would
naturally be ragarded as true" without formalizing this notion. Mine was
just an Idea, no matter if nobody here like it...


0
1/12/2005 8:48:58 PM
Dave Seaman wrote:
> On Wed, 12 Jan 2005 13:52:53 -0500, Ron Sperber wrote:
> 
>>Torkel Franzen wrote:
>>
>>>Ron Sperber <ronsperber@optonline.net> writes:
>>>
>>>
>>>
>>>>In other words, ZFC is  a model.
>>>
>>>
>>>  In mathematical terminology, ZFC is not a model but a formal theory.
>>>What does the above statement mean?
>>
>>How is it not a model for set theory? In other words, ZFC describes 
>>axioms that sets obey. If we start with these axioms, one can prove 
>>somethings and disprove some things, and fail to be able to do either 
>>with some others.
> 
> 
> Do the axioms for a group form a model for group theory?  Which of the
> axioms serves as the identity in this model?
> 
> 
No, but the axioms for a group allow us to prove things in general for 
groups. If I'm using "model" in some non-standard way, I apologize. What 
I meant was that one can start with some set of axioms (e.g. ZFC) and 
then either prove things or disprove them or state that they can be 
neither proved nor disproved. The "truth" of ZFC never enters into it. 
One can only ask if it is consistent. Of course there is a question of 
usefulness as well, but so far ZFC seems to be useful.
0
1/12/2005 9:17:27 PM
On Wed, 12 Jan 2005 16:17:27 -0500, Ron Sperber wrote:
> Dave Seaman wrote:
>> On Wed, 12 Jan 2005 13:52:53 -0500, Ron Sperber wrote:
>> 
>>>Torkel Franzen wrote:
>>>
>>>>Ron Sperber <ronsperber@optonline.net> writes:
>>>>
>>>>
>>>>
>>>>>In other words, ZFC is  a model.
>>>>
>>>>
>>>>  In mathematical terminology, ZFC is not a model but a formal theory.
>>>>What does the above statement mean?
>>>
>>>How is it not a model for set theory? In other words, ZFC describes 
>>>axioms that sets obey. If we start with these axioms, one can prove 
>>>somethings and disprove some things, and fail to be able to do either 
>>>with some others.
>> 
>> 
>> Do the axioms for a group form a model for group theory?  Which of the
>> axioms serves as the identity in this model?
>> 
>> 
> No, but the axioms for a group allow us to prove things in general for 
> groups. If I'm using "model" in some non-standard way, I apologize. What 
> I meant was that one can start with some set of axioms (e.g. ZFC) and 
> then either prove things or disprove them or state that they can be 
> neither proved nor disproved. The "truth" of ZFC never enters into it. 
> One can only ask if it is consistent. Of course there is a question of 
> usefulness as well, but so far ZFC seems to be useful.

ZFC is a collection of axioms.  The empty set, for example, is not an
axiom and therefore is not a member of ZFC.  There is an axiom (in some
versions of ZFC) saying that the empty set exists, but that axiom is not
the same thing as the set itself.  A model of ZFC would include some
actual representation of the empty set.  For example, a sufficiently
large cardinal might serve as a model of ZFC, but the elements of that
model are sets, not axioms.

Also, I'm not sure what you mean by "consistent"?  The usual meaning is
that a system is consistent if it has a model.


-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
0
dseaman (1174)
1/12/2005 9:42:18 PM
In article <cs3819$klf$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>With that caveat, I don't know of any real physical theory that permits the 
>computation of non-recursive functions.

O.K., thanks for the clarification; I did indeed not fully get your meaning
the first time around, sorry.

However, I claim that the physical CT thesis is not really relevant, because
the real question is, what justifies the assumption that we acquire 
mathematical knowledge by *computation*?

You address this elsewhere in this thread, so I won't say more in this
message; just wanted to "close out" this branch of the thread regarding
the CT thesis.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 9:43:42 PM
In article <ExgFd.622$BU6.397@fe08.lga>,
Ron Sperber  <ronsperber@optonline.net> wrote:
>I meant was that one can start with some set of axioms (e.g. ZFC) and 
>then either prove things or disprove them or state that they can be 
>neither proved nor disproved. The "truth" of ZFC never enters into it. 
>One can only ask if it is consistent. Of course there is a question of 
>usefulness as well, but so far ZFC seems to be useful.

You're really answering a different question from the one that is
being discussed.  The question was: Are there permanently unknowable
[mathematical] truths?  This tacitly assumes that there are *some*
truths, and that some truths are knowable.  Do you maintain that
there are no truths?  You can't say, "The `truth' [of ZFC] never
enters into it," because here I'm stipulating that "it" is the question,
"Are there permanently unknowable truths?" and the word "truth" is
forced on us by the topic of discussion.

If you maintain that there are no truths, or refuse to address the question
of truths, then presumably you a fortiori maintain that there can't be any
truths that additionally have the property of being permanently unknowable,
so you're on my side of the debate, and I have no further argument with you.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 9:49:33 PM
In article <cs3ep502ec9@drn.newsguy.com>,
Daryl McCullough <daryl@atc-nycorp.com> wrote:
>Then forget reflection. I'm making the empirical conjecture
>that for mathematical problems arising in fields outside of
>mathematical logic and set theory, if they are humanly solvable
>at all, then the solution can be formulated as a proof in ZFC.

I think that it's probably not so easy to "forget reflection" so glibly.
Harvey Friedman's results suggest that problems arising in ordinary
mathematics may well require large cardinal axioms.  He has examples
of Pi^0_1 statements that are fairly "natural-looking," making no
explicit or thinly-disguised reference to logic, that are provable
using (say) ZFC+"Mahlo cardinals" and that themselves imply the
consistency of ZFC+"cardinals slightly smaller than Mahlo."  So if
we're trying to peer into the future, reflection may be forced on us.

And once we open the door to reflection, it is rather compelling to
regard the addition of new axioms as an open-ended procedure with no
sharp boundary.  Every time you try to close the lid (e.g., with ZFC),
there will be the temptation for something else (e.g., Con(ZFC)) to
be added to what constitutes "knowledge."
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 10:01:59 PM
Torkel Franzen wrote:
> Ralph Hartley <hartley@aic.nrl.navy.mil> writes:
> 
>>Given a set of 100 facts that I cannot determine the truth of, I can *hold* 
>>the correct evaluation of about 50 of them by guessing at random. That does 
>>not mean I *know* the truth of any of them.
> 
>   Precisely! So what distinction between knowing and mere holding do
> you have in mind with reference to mathematical statements?

Knowledge is usually defined as belief that is *both* justified, and correct.

Do you think there is any difference between knowledge and correct belief?

If not, then I don't think you will agree that there are true statements 
that cannot be known.

Ralph Hartley
0
hartley (156)
1/12/2005 10:02:20 PM
In article <QP3Fd.57570$8l.50064@pd7tw1no>,
namducnguyen  <namducnguyen@shaw.ca> wrote:
>On my first question:
> >>1) As far as FOL is concerned, what is the definition of "_permanently_
> >>unknownable truth"?
>you gave a very "strong" answer:
> >There isn't one.

Yes, I meant that there is no standard definition of such a term,
especially not in the context of FOL.

>On my 2nd question:
> >>2) What does Godel's work has to do with "permanently unknownable 
>truths"?
>the answer "Nothing directly" sorts of implies that  _indirectly_ there 
>might be some
>relationship between Godel's work and the "permanently unknownable 
>truths"

Yes, but I was just being cautious; since your 2nd question was more
open-ended and philosophical, I didn't want to sound like I could
*rule out* the possibility of a relationship.  But that doesn't mean
I have any clear relationship between the two in mind.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 10:05:28 PM
In article <cs46gl$l0f$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>Knowledge is usually defined as belief that is *both* justified, and correct.

Not since Gettier (Google "Gettier counterexamples" if the name is
unfamiliar to you).
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 10:08:02 PM
In article <41e59b6d$0$564$b45e6eb0@senator-bedfellow.mit.edu>,
 tchow@lsa.umich.edu wrote:
[snip]

>The question was: Are there permanently unknowable
>[mathematical] truths?  This tacitly assumes that there are *some*
>truths, and that some truths are knowable.  Do you maintain that
>there are no truths?  You can't say, "The `truth' [of ZFC] never
>enters into it," because here I'm stipulating that "it" is the question,
>"Are there permanently unknowable truths?" and the word "truth" is
>forced on us by the topic of discussion.

Would you consider statements of the form "proposition P has a proof in 
theory T" to be candidate mathematical truths?  If so, then every formal 
proof of anything also represents a computationally-verifiable mathematical 
truth, without any ambiguity or additional philosphical committment.

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/12/2005 10:21:42 PM
tchow@lsa.umich.edu says...
>
>In article <cs3ep502ec9@drn.newsguy.com>,
>Daryl McCullough <daryl@atc-nycorp.com> wrote:
>>Then forget reflection. I'm making the empirical conjecture
>>that for mathematical problems arising in fields outside of
>>mathematical logic and set theory, if they are humanly solvable
>>at all, then the solution can be formulated as a proof in ZFC.
>
>I think that it's probably not so easy to "forget reflection" so glibly.
>Harvey Friedman's results suggest that problems arising in ordinary
>mathematics may well require large cardinal axioms.  He has examples
>of Pi^0_1 statements that are fairly "natural-looking," making no
>explicit or thinly-disguised reference to logic, that are provable
>using (say) ZFC+"Mahlo cardinals" and that themselves imply the
>consistency of ZFC+"cardinals slightly smaller than Mahlo."  So if
>we're trying to peer into the future, reflection may be forced on us.

If that's true, I'm in trouble, because I've tried to formulate the most general
notion of what can be considered a "reflection principle", but it's very hard to
pin it down.

>And once we open the door to reflection, it is rather compelling to
>regard the addition of new axioms as an open-ended procedure with no
>sharp boundary.  Every time you try to close the lid (e.g., with ZFC),
>there will be the temptation for something else (e.g., Con(ZFC)) to
>be added to what constitutes "knowledge."

I agree, it's hard to know exactly where to stop.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/12/2005 10:24:36 PM
In article <cs3b0n$kmq$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>No. Perhaps "effective" is not exactly the word I intended to use? I 
>intended to include *partial* recursive functions. Does "effective" have 
>such a precice definition that it excludes them?

O.K., fair enough.  I wasn't sure to what extent you were defending the
earlier posters in this thread, who seemed to think that the halting problem
already shows that there are permanently unknowable truths---I took them to
regard partial recursive functions as being not good enough to yield
knowledge.  So forget that example.

But in any case, the real issue again is your assumption that
mathematicians use an effective procedure to determine mathematical truth.
As I've argued elsewhere in this thread, CT theses of whatever flavor
don't yield this assumption; CT theses will only say that if mathematicians
are indeed using an effective procedure to find, or are *computing*,
mathematical truths, then the theory of [partial] recursive functions
applies and lets us deduce further conclusions.  But it doesn't say
whether mathematicians are indeed using an effective procedure.  Even
if the universe is computable in some sense, it doesn't follow that
everything that takes place inside the universe is an effective procedure.
For example, I know of nobody---atheist or believer---who thinks that
theological doctrines are generated by an effective procedure.

You've argued that if something is obtained by a procedure that is not
effective, then it's not *knowledge*.  Perhaps that's true, but the
the corpus of what is *generally called* "mathematical knowledge" is
not generated by an effective procedure in any obvious way.  "Every
vector space has a basis" is generally considered to be a mathematical
truth.  It can be proved using various axioms, including the axiom of
choice.  But how did we come to accept the axiom of choice?  There were
heated arguments, and through a complex sociological process, the axiom
of choice won out.  Was this an effective procedure?  Was it even objective?
What's to stop some similar mess from happening again---say, with the axiom
of projective determinacy, which Woodin and others have been advocating?
If these things aren't knowledge, or aren't mathematical truths, then
your assumption comes at the cost of discarding a lot of what most people
consider to be mathematical truth.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/12/2005 10:31:15 PM
tchow@lsa.umich.edu wrote:
> In fact, do we know *any* of the axioms of ZFC?  Or of PA?  Or of any other
> axiomatic system you might write down?

That's why when you want to be picky we say "such and such is true in ZFC".

When we want to me *more* picky we say "thus and so is a theorem in ZFC".

> Maybe you can get everyone to agree
> on some axioms, but is agreement really enough to yield *knowledge*?  Surely
> not.  We could all agree on something that is false, right?

Happens all the time.

> We could take the extreme skeptical view, that we cannot know anything.
> Then to deduce the existence of permanently unknowable truths, all we
> need is the existence of *some* truth  (which perforce is permanently
> unknowable).  But I doubt that you are advocating such an extreme position.

That's what I meant when I said that the idea that we know *anything* is a 
little white lie. Of course, we all tell it all the time, and it is close 
enough to being true (in some sense, that I will not, or cannot explain), 
that we mostly get away with it.

> Most likely you think we do know some things---"finitary" statements
> perhaps, such as those formalizable in primitive recursive arithmetic
> (PRA).  Do you think that PRA exhausts mathematical knowledge?

Actually, I have no problem with ZFC.

I have no problem with the CH or any other axioms, as long as it is clear 
(explicitly or by default) which you are using.

I don't know if there is any sense in which AC can be said to *be* true or 
false, but I'm pretty sure it is true in ZFC :-).

I am a bit more ambivalent toward statements like "this polynomial equation 
has no integer solution". I would tend to think that such a statement 
*ought* to be either true or false. Period. I wouldn't make me feel better 
for most mathematicians to agree that there should be an axiom saying that 
it is true.

> But getting back to the
> question of "permanently unknowable truths," it's not clear to me how
> you would be able to establish their existence.  PRA isn't strong enough
> to prove Goedel's theorems, for example.

Modulo the choice of example, there are weaker systems in which there are 
no permanently unknowable truths.

One might guess that there would be fewer things you can't know in a 
stronger system, but it is the other way around.

Ralph Hartley
0
hartley (156)
1/12/2005 10:35:14 PM
Ron Sperber wrote:
> Torkel Franzen wrote:
>> Ron Sperber <ronsperber@optonline.net> writes:
>>> In other words, ZFC is  a model.
>>   In mathematical terminology, ZFC is not a model but a formal theory.
>> What does the above statement mean?
> 
> How is it not a model for set theory? In other words, ZFC describes 
> axioms that sets obey. If we start with these axioms, one can prove 
> somethings and disprove some things, and fail to be able to do either 
> with some others.

Two different senses of the word "model".

Mathematicians use the word in a perverse way. In their sense, sets are a 
model for ZFC.

In the *ordinary* sense of the word it is the other way around.

It causes less confusion than one would expect.

Ralph Hartley
0
hartley (156)
1/12/2005 10:41:47 PM
"Barb Knox" <see@sig.below> ha scritto

> >The question was: Are there permanently unknowable
> >[mathematical] truths?  This tacitly assumes that there are *some*
> >truths, and that some truths are knowable.  Do you maintain that
> >there are no truths?  You can't say, "The `truth' [of ZFC] never
> >enters into it," because here I'm stipulating that "it" is the question,
> >"Are there permanently unknowable truths?" and the word "truth" is
> >forced on us by the topic of discussion.
>
> Would you consider statements of the form "proposition P has a proof in
> theory T" to be candidate mathematical truths?  If so, then every formal
> proof of anything also represents a computationally-verifiable
mathematical
> truth, without any ambiguity or additional philosphical committment.

Even if the existence of the proof is proved by some nonconstructive
methods?


0
1/12/2005 10:42:40 PM
tchow@lsa.umich.edu wrote:
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> 
>>Knowledge is usually defined as belief that is *both* justified, and correct.
> 
> Not since Gettier

OK, but being justified and true are still *necessary* for a belief to be 
knowledge.

I suspect that it is *still* usually defined that way, even though that 
definition is not without problems.

>  (Google "Gettier counterexamples" if the name is unfamiliar to you).

It wasn't. Thanks.

Ralph Hartley
0
hartley (156)
1/12/2005 11:09:02 PM
In article <AJhFd.399207$b5.19403050@news3.tin.it>,
 "LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> wrote:

>"Barb Knox" <see@sig.below> ha scritto
>
>> >The question was: Are there permanently unknowable
>> >[mathematical] truths?  This tacitly assumes that there are *some*
>> >truths, and that some truths are knowable.  Do you maintain that
>> >there are no truths?  You can't say, "The `truth' [of ZFC] never
>> >enters into it," because here I'm stipulating that "it" is the question,
>> >"Are there permanently unknowable truths?" and the word "truth" is
>> >forced on us by the topic of discussion.
>>
>> Would you consider statements of the form "proposition P has a proof in
>> theory T" to be candidate mathematical truths?  If so, then every formal
>> proof of anything also represents a computationally-verifiable
>mathematical
>> truth, without any ambiguity or additional philosphical committment.
>
>Even if the existence of the proof is proved by some nonconstructive
>methods?

If it's a *formal* non-constructive proof (in some system TT) of the 
existence of a proof in T of P, then yes.  The importance of it being a 
formal proof is that it can be algorithmically verified, not needing any 
additional philosophical assumptions.  Note in this case that the "truth" is 
TT |- (T |- P), not T |- P itself.

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/12/2005 11:17:20 PM
"Barb Knox" <see@sig.below> ha scritto

> >> Would you consider statements of the form "proposition P has a proof in
> >> theory T" to be candidate mathematical truths?  If so, then every
formal
> >> proof of anything also represents a computationally-verifiable
> >mathematical
> >> truth, without any ambiguity or additional philosphical committment.
> >
> >Even if the existence of the proof is proved by some nonconstructive
> >methods?
>
> If it's a *formal* non-constructive proof (in some system TT) of the
> existence of a proof in T of P, then yes.  The importance of it being a
> formal proof is that it can be algorithmically verified, not needing any
> additional philosophical assumptions.  Note in this case that the "truth"
is
> TT |- (T |- P), not T |- P itself.

Uhm... and what if I prove T |- P (in TT) using things like axiom of choice,
transfinite induction or existence of some large cardinal?
If we accept this proof as valid why don't we consider "mathematical truth"
the whole set theory without restricting the notion of "mathematical truth"
to statements of the form "proposition P has a proof in theory T"?


0
1/12/2005 11:37:48 PM
In article <gxiFd.399412$b5.19413518@news3.tin.it>,
 "LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> wrote:

>"Barb Knox" <see@sig.below> ha scritto
>
>> >> Would you consider statements of the form "proposition P has a proof in
>> >> theory T" to be candidate mathematical truths?  If so, then every
>formal
>> >> proof of anything also represents a computationally-verifiable
>> >mathematical
>> >> truth, without any ambiguity or additional philosphical committment.
>> >
>> >Even if the existence of the proof is proved by some nonconstructive
>> >methods?
>>
>> If it's a *formal* non-constructive proof (in some system TT) of the
>> existence of a proof in T of P, then yes.  The importance of it being a
>> formal proof is that it can be algorithmically verified, not needing any
>> additional philosophical assumptions.  Note in this case that the "truth"
>is
>> TT |- (T |- P), not T |- P itself.
>
>Uhm... and what if I prove T |- P (in TT) using things like axiom of choice,
>transfinite induction or existence of some large cardinal?
>If we accept this proof as valid why don't we consider "mathematical truth"
>the whole set theory without restricting the notion of "mathematical truth"
>to statements of the form "proposition P has a proof in theory T"?

I'm not claiming that statements of the form "T |- P" exhaust all possible 
sorts of mathematical truth, but that they are a subset.  I am not 
addressing whether or not (e.g.) AC is "true" in some other sense.

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/12/2005 11:44:01 PM
tchow@lsa.umich.edu wrote in message
<41e55b00$0$576$b45e6eb0@senator-bedfellow.mit.edu>...

>Yes, sorry.  I realized my gaffe just after posting.  Seems my anti-spam
>mechanisms also prevent me from canceling my own messages.

For a talk I gave at Bristol last week, I sent the abstract to the
organizer. He pointed out that I'd mispelled the word "dalmatian" in my main
example! Reading the overheads the night before, I noticed that I'd
expressed "R is a many-one relation" the wrong way round.

>More generally, (a version of) Goedel's incompleteness theorem is provable
>in RCA_0, and Goedel's completeness theorem is provable in WKL_0, and WKL_0
>is conservative over PRA for Pi^0_2 sentences.  I think Goedel's theorems
>can be formulated as Pi^0_2 sentences.

Yes, WKL_0 is a bit nicer than ISigma_1, because you can talk about certain
sets of numbers, and the reasoning would be closer to informal reasoning.

Both theorems have the form "forall x A(x) -> forall y B(y)" which prenex
out as Pi_2 sentences.

More detail. The 1st Incompleteness Theorem (for fixed recursively
axiomatized T extending Q, with appropriate goedel coding, etc.) has the
form "forall n(n is not a proof of 0=1 in T) -> forall m(m is not a proof of
G in T)".
Here I'm ignoring that there isn't a proof of ~G either.
This is equivalent to Con(T)->G, and becomes "forall x, exists y(~Proof_T(y,
[0=1]) -> ~Proof_T(x, [G]))", which is equivalent to the Pi_2 sentence:

      (*) forall x, exists y(Proof_T(x, [G]) -> Proof_T(y, [0=1]))

It follows that there is a Turing machine M_T which converts any proof of G
in T to a proof of 0=1 in T.
Similarly, if G is a Goedel-Rosser sentence, then there will be a machine
which converts a proof of ~G in T to a proof of 0=1.

2nd incompleteness theorem has the form "forall x, exists y(~Proof_T(y,
[0=1]) -> ~Proof_T(x, [Con(T)]))" which is equivalent to the Pi_2 sentence:

     (**) forall x, exists y(Proof_T(x, [Con(T)]) -> Proof_T(y, [0=1]))

So, there's a machine that converts any proof of Con(T) in T to a proof of
0=1 in T.

As you say, (*) and (**) will be provable in WKL_0.

--- Jeff


0
ketland (18)
1/13/2005 1:17:37 AM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> writes:

> In the same way I wanted to say something about some "statements that would
> naturally be ragarded as true" without formalizing this notion.

  G�del regarded it as a kind of miracle that the informal concept of
mechanical computability could in fact be captured in a formal
definition. Provability is much more problematic. There was no
disagreement in mathematics over particular algorithms - it was clear
to everybody that Sturm's algorithm was an algorithm, that Euclid's
algorithm was an algorithm, and so on. In the case of provability,
there is disagreement over what constitutes a proof, and there is
a distinction between more or less conclusive or convincing proofs.





0
torkel (478)
1/13/2005 3:49:16 AM
Ron Sperber <ronsperber@optonline.net> writes:

> What I meant was that one can start with some set of axioms (e.g. ZFC) and 
> then either prove things or disprove them or state that they can be 
> neither proved nor disproved. The "truth" of ZFC never enters into it. 
> One can only ask if it is consistent. 

  This "one can only ask" is completely arbitrary. Why shouldn't we
ask, for example, whether provability in ZFC of "there are infinitely
many twin primes" guarantees that there are infinitely many twin
primes?


0
torkel (478)
1/13/2005 3:51:09 AM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> Knowledge is usually defined as belief that is *both* justified, and
> correct.

  And the question at issue is what amounts to justification in
mathematics. To the extent that this is ill-defined, or a matter of
personal inclination, we have formulated no theoretical question
concerning the existence of truths that are not recognizable, etc.

0
torkel (478)
1/13/2005 3:54:07 AM
tchow@lsa.umich.edu wrote:
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> 
> I wasn't sure to what extent you were defending the
> earlier posters in this thread

To be honest, neither was I.

I have spent way more time on this thread than I should have, and learned 
more than I expected. But I can't continue much more.

> But in any case, the real issue again is your assumption that
> mathematicians use an effective procedure to determine mathematical truth.
> As I've argued elsewhere in this thread, CT theses of whatever flavor
> don't yield this assumption; CT theses will only say that if mathematicians
> are indeed using an effective procedure to find, or are *computing*,
> mathematical truths, then the theory of [partial] recursive functions
> applies and lets us deduce further conclusions.  But it doesn't say
> whether mathematicians are indeed using an effective procedure.

This argument is not without merit. I don't *think* I totally agree, but I 
don't have time to decide for sure, or to explain my problems with it much 
more than I already have.

It may not be enough, for your argument, for the procedure to not be 
effective, it needs to be *better* than any effective procedure. In 
particular it needs to be both sound and complete.

Otherwise you will come to "know" false things, or there are true things 
you will never know.

If a mathematician in a particular universe can correctly answer all 
members of a class of questions (by any means, effective or not), and there 
is any recursive process to determine what she has concluded, and my 
version of the physical CT thesis holds for that universe, then the class 
of questions is recursive.

> I know of nobody---atheist or believer---who thinks that
> theological doctrines are generated by an effective procedure.

I'm not sure I would go so far as to say that I *think* that, but I see no 
reason to conclude that it isn't. There are effective procedures for 
producing nonsense.

Some believers consider reference to a particular text to be the first, 
last, and only way to obtain truth. Looking up the answer in a book seems 
pretty effective to me.

> You've argued that if something is obtained by a procedure that is not
> effective, then it's not *knowledge*.  Perhaps that's true, but the
> the corpus of what is *generally called* "mathematical knowledge" is
> not generated by an effective procedure in any obvious way.  "Every
> vector space has a basis" is generally considered to be a mathematical
> truth.  It can be proved using various axioms, including the axiom of
> choice.  But how did we come to accept the axiom of choice?

I'm not sure I would call "Every vector space has a basis" a mathematical 
truth if that is what you really meant when you said it. I think I *would* 
call "In ZFC every vector space has a basis" a mathematical truth.

But you are presumably a mathematician, and when mathematicians make 
unqualified statements, with no other context, they usually *mean* "In ZFC 
....".

In a sense, mathematical axioms are not knowledge because they are not 
statements that can be true or false in an absolute sense. They can be 
viewed as being more like definitions.

The axioms of group theory are not facts that are "known", they describe 
what we *mean* when we talk about a group.

There was once quite a bit of fuss over the truth of the Parallel 
Postulate. Nowadays, we would call it a property of a space. It isn't true 
or false in an absolute sense. Some spaces have it and some don't.

Similarly, accepting AC can be viewed as being more specific about what you 
mean by "sets".

I am not sure I am willing to follow this line of reasoning to its ultimate 
destination. I like to think I know what I mean by *the* integers, and 
there are some statements that I might have trouble viewing as "a matter of 
definition", even though they are independent of all the axioms.

I imagine people used to feel that way about points and lines.

I really am quite fond of that "little white lie".

> There were heated arguments, and through a complex sociological process, the axiom
> of choice won out.  Was this an effective procedure?  Was it even objective?

Which is exactly what one would expect if it was a matter of definition, 
not of truth. Truth isn't normally considered a matter of consensus, and is 
not considered negotiable, but definitions are.

Definitions can be produced by an effective procedure or not, because there 
is no need for them to be objective.

There are good definitions and bad ones, but it is mostly a matter of 
utility. Asking if a definition is *true* is nonsense (we can, and should, 
ask if a definition is consistent).

> What's to stop some similar mess from happening again---say, with the axiom
> of projective determinacy, which Woodin and others have been advocating?

It most certainly will, if not in that case, then in some other.

> If these things aren't knowledge, or aren't mathematical truths, then
> your assumption comes at the cost of discarding a lot of what most people
> consider to be mathematical truth.

In the case of set theory, that would be less than one page of text. 
Including definitions might require a small font.

The axioms themselves are a very small part of mathematics. Most (one could 
argue all) mathematical knowledge is in the theorems.

One way to evaluate a mathematical theory is to look at its "theorem to 
axiom ratio". All else being equal, bigger is better.

A big bunch of complex axioms, from which little additional can be proven 
is barely mathematics. We like small a set of simple axioms with an 
enormous number and diversity of theorems (e.g. set theory).

"Good" theories are more useful. The small set of axioms mean they apply 
more often, and the large set of theorems mean you get a lot of answers.

I suspect that one reason that AC is accepted is that it is simple and 
produces many diverse theorems.

It is unreasonable to expect to get an answer to *every* question. Some of 
the *best* theories allow one to express questions that they cannot answer.

You can always *define* answers to questions that your axioms don't answer, 
but it is unclear in what sense that is the same as knowing the answer, and 
there in no sure (or even effective) way to avoid making inconsistent 
assumptions.

If you keep adding axioms, without proving then consistent, you will surely 
add an inconsistent one eventually.

Some statements cannot be proven "safe" to use as axioms (even if they 
are). (The safe axioms are the complement of an r.e. set) If you only add 
safe axioms, there are some statements you will never decide.

AC is a not the best example, since it *was* proven safe.

Ralph Hartley
0
hartley (156)
1/13/2005 6:35:39 PM
In article <cs47tm$umk$1@lust.ihug.co.nz>, Barb Knox  <see@sig.below> wrote:
>Would you consider statements of the form "proposition P has a proof in 
>theory T" to be candidate mathematical truths?  If so, then every formal 
>proof of anything also represents a computationally-verifiable mathematical 
>truth, without any ambiguity or additional philosphical committment.

I do consider such statements to be candidate mathematical truths.  What I
don't accept, and what you said you also don't accept, is that these are the
*only* candidate mathematical truths.  In the absence of such an assumption,
what justifies the belief that there are permanently unknowable mathematical
truths?  Nothing that I can see.  The undecidability of the halting problem
does not furnish any plausible candidates.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/14/2005 3:19:38 AM
In article <cs4834030ho@drn.newsguy.com>,
Daryl McCullough <daryl@atc-nycorp.com> wrote:
>If that's true, I'm in trouble, because I've tried to formulate the
>most general notion of what can be considered a "reflection principle",
>but it's very hard to pin it down.

On this point, you and Torkel Franzen are in agreement.  See his recently
published book "Inexhaustibility."

I'm still not entirely sure what my opinion is on this subject, but I
think I'm settling down to something like this.  The question of whether
the set of all eventually knowable mathematical truths is r.e. is, I
believe, open---that is, if the question can even be made precise enough
to be meaningful.  Attempts of people like Penrose to show "definitively"
that this set is *not* r.e. fail, for various reasons, one of the simplest
being that it is always possible to "dig in one's heels" and declare some
system (PRA, ZFC, whatever) to be "the" answer, and flatly refuse to
entertain reflection.  But on the flip side, digging in one's heels is
not very satisfying because the choice of system seems rather arbitrary,
plus reflection is very tempting.  The situation looks like a stalemate,
unless someone can find some new philosophical insight in this rather
well-tilled area.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/14/2005 3:30:24 AM

tchow@lsa.umich.edu wrote:

>In article <cs47tm$umk$1@lust.ihug.co.nz>, Barb Knox  <see@sig.below> wrote:
>  
>
>>Would you consider statements of the form "proposition P has a proof in 
>>theory T" to be candidate mathematical truths?  If so, then every formal 
>>proof of anything also represents a computationally-verifiable mathematical 
>>truth, without any ambiguity or additional philosphical committment.
>>    
>>
>
>I do consider such statements to be candidate mathematical truths.  What I
>don't accept, and what you said you also don't accept, is that these are the
>*only* candidate mathematical truths.  In the absence of such an assumption,
>what justifies the belief that there are permanently unknowable mathematical
>truths?  Nothing that I can see.  The undecidability of the halting problem
>does not furnish any plausible candidates.
>  
>

I wonder, though, what would happen if P is a proposition in theory T 
that we can't never
know if there is a proof in T (notwithstanding that it must either have 
or have not
a proof in T)? Could "proposition P has a proof in theory T" be a 
candidate for an
_eternal_ [mathematical] truth?
0
1/14/2005 3:40:35 AM
tchow@lsa.umich.edu writes:

> The question of whether
> the set of all eventually knowable mathematical truths is r.e. is, I
> believe, open---that is, if the question can even be made precise enough
> to be meaningful.

  Nobody has come close to formulating this as a meaningful question.
0
torkel (478)
1/14/2005 3:52:08 AM
In article <cs6ep3$lso$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>But you are presumably a mathematician, and when mathematicians make 
>unqualified statements, with no other context, they usually *mean* "In ZFC 
>...".

As a matter of sociological fact, this is definitely false.  For a trivial
counterexample, consider:

  (*)  "sqrt(2) is irrational" is provable in ZFC.

When mathematicians assert (*), they mean (*).  They don't mean

  (**) `"sqrt(2) is irrational" is provable in ZFC' is provable in ZFC.

They couldn't, obviously, because this would instantly lead to an infinite
regress.  Even you seem comfortable with asserting (*) flat out; that is,
it isn't a white lie or a shorthand for something like (**)---it is
meaningful on its own.  So we have at least one class of examples of
mathematical statements that are meaningful on their own.  Mathematicians
typically include "sqrt(2) is irrational" and "every differentiable function
is continuous" and so forth among the mathematical statements that are
meaningful on their own, and that are *true* in an absolute sense, just
as (*) is true in an absolute sense.

Now maybe they are wrong to do so, but that's how they are.

There's another way to see that ZFC doesn't, in practice, have the status
that you assign it.  Suppose that someone were to find a contradiction
in ZFC.  Would this make any difference to mathematics?  It would
depend on the specific contradiction, but in general, it wouldn't make
any difference.  Logicians would just pick some other foundation for
mathematics with no known contradiction.  ZFC is way too strong for most
of ordinary mathematics anyway.  All the theorems in the books would
remain intact, except for the few that were affected by the specific
contradiction.  What would you say then?  That in this new situation,
"sqrt(2) is irrational" no longer means "`sqrt(2) is irrational' is
provable in ZFC" but now means "`sqrt(2) is irrational' is provable in X,"
where X is the new foundation?  Given that the usual proof of sqrt(2)'s
irrationality is left unchanged by the discovered contradiction in ZFC,
it's a little bizarre to think that its meaning has changed.  It is surely
more plausible that "sqrt(2) is irrational" meant, and still would mean,
"sqrt(2) is irrational" and not "`sqrt(2) is irrational' is provable
in something-or-other."  And that's the way most mathematicians view it.
The familiar proof that sqrt(2) is irrational consists of a sequence of
meaningful statements that we can read and understand and that leads us
to accept the truth of "sqrt(2) is irrational"; mimicking this proof
formally in this or that formal system does not yield the "true meaning"
of the statements in question.  In fact, it's almost the other way around;
we only accept (as a candidate for foundations) formal systems that
faithfully mimic what we *already* recognize to be correct reasoning.
Where do you think ZFC came from in the first place?

This is not to say that your point of view, which is roughly some kind of
finitism or formalism, is untenable.  But it involves a whole host of
assumptions, many of which don't agree with how mathematicians actually
work with and view mathematical statements.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/14/2005 3:59:01 AM
In article <cs4adn$l1b$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
[Re: knowledge as true justified belief]
>I suspect that it is *still* usually defined that way, even though that 
>definition is not without problems.

I would be surprised to find that definition taken for granted in any
professional philosophical paper (or even textbook on epistemology)
since Gettier, except perhaps for papers specifically trying to argue
that Gettier's (and others') objections were unfounded.

As the old joke goes, if you ask a question of three philosophers,
you'll get five different answers.  This certainly applies to the
question, "What is the definition of `knowledge'?"  Axiomatizations
of the concept of knowledge are a dime a dozen.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/14/2005 4:14:37 AM
In article <TaHFd.70924$8l.16323@pd7tw1no>,
namducnguyen  <namducnguyen@shaw.ca> wrote:
>I wonder, though, what would happen if P is a proposition in theory T 
>that we can't never
>know if there is a proof in T (notwithstanding that it must either have 
>or have not
>a proof in T)? Could "proposition P has a proof in theory T" be a 
>candidate for an
>_eternal_ [mathematical] truth?

If it is permanently unknowable whether P is provable in T, then `P is
unprovable in T' would be a permanently unknowable truth, yes.  But are
there plausible candidates for such a P and T?
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/14/2005 4:18:51 AM
namducnguyen wrote in message ...

>I wonder, though, what would happen if P is a proposition in theory T
>that we can't never
>know if there is a proof in T (notwithstanding that it must either have
>or have not
>a proof in T)? Could "proposition P has a proof in theory T" be a
>candidate for an
>_eternal_ [mathematical] truth?

Can there arise a situation where T implies P, but we "can't [n]ever know if
there is a proof in T"?
Possibly.

There are cases of P such that T proves P, but the shortest proof of P in T
is so long that merely by trying to find a proof in T we cannot (easily)
succeed, and in some cases, we humans simply could never in practice
succeed, given obvious limitations on our time and memory resources.

The reason is that proofs (in the abstract sense---an arbitrary long
sequence or tree of some sort) of formulas, even logically valid formulas,
can be astronomically long.

An example of such a valid formula of FOL whose shortest proof must be huge
was given by George Boolos in 1987. Consider the formula, Boolos(4, 4):

   forall x, y [fxa = sa & fasx = ssfax & fsxsy = fxfsxy &
   Na & (Nx -> Nsx)] -> Nfssssassssa

where f is a binary function symbol, s is a unary function symbol and N is a
monadic predicate symbol.

The formula Boolos(4, 4) is *valid*, which means that it is true in all
models, and is of course provable in any complete deductive system for FOL.
However, Boolos estimates that the shortest proof of Boolos(4, 4) in a
"Mates-style deductive system" for first-order logic would have length of
the order of an exponential stack of 65,536 2's.
I.e., 2^{2^{2^{............2}............}}}, which is astronomically vast.

The idea behind Boolos's formula here is based on a rapidly growing
recursive function (a la Ackermann), and the fact that any proof would have
to use instances Na -> Nsa, Nsa -> Nssa, etc., the appropriate number of
times, up to the value of f(4, 4), and fssssassssa has to be represented as
ssss........sa, with the above number of s's.
Boolos(4, 4) *is* valid, but you cannot demonstrate this by a purely logical
derivation (unless you arbitrarily re-axiomatize FOL in an ad hoc manner,
e.g., by taking instances of Boolos(n, m) amongst the axioms).

First noted by Kurt Goedel in 1936, this "speed-up" phenomenon occurs
generically for certain pairs of theories, T+ and T. E.g., in order to prove
that T implies P, you may have to use a richer mathematical theory T+ in
order to get a proof that has less than, say, 10,000 symbols (~3 pages).
E.g., T might be PA, and T+ might be ZF, while an independent argument might
show that the shortest proof of P in T has > N symbols, with N very large.

Some formalists or nominalists think that all mathematical knowledge can be
reduced to "validity facts", of the conditional or implicational form "A
implies B". Simple examples are "PA implies that there are infinitely many
primes" or "Z set theory implies the Intermediate Value Theorem".
But it follows from the above discussion that you may still need to assume a
certain more powerful mathematical theory M in order to *prove* that A
implies B (in some cases, and where A really does imply B).
If you do so, why should you trust the more powerful theory M? Even if M
implies "A implies B", how can you justify concluding that A *does* imply B?
You may conclude that A implies B from this sort of proof only if you
already agree at least that whenever your mathematical theory M implies a
validity-claim (i.e., of the form "A implies B"), then this validity-claim
is true.
In other words, you need the Val-Reflection Principle for your theory M:

   If M implies "A is valid", then A is valid.

(I would be happy to assume this for Z set theory. If I can prove using
set-theoretic methods that a formula A is valid, then I would conclude that
A is, in fact, valid.)

Is the formula Boolos(4, 4) really valid? An eternal mathematical truth,
even for a formalist? Presumably so. But how would one prove that Boolos(4,
4) is valid if one disbelieved the mathematics that appears to be
indispensable to prove this? These feasibility considerations show that
formalist scruples will prevent formalists from recognizing some facts---of
precisely the sort they claim are mathematical facts---namely certain facts
of the form "A implies B" or "A is valid".

Let me add that I do not have a clear idea how to answer the question
whether there are "permanently unknowable truths". Because I am realist, I
suspect that there are, for the simple reason that mathematical reality is
not our creation and mathematical reality could easily transcend our
epistemic capacities somehow. But that's an inconclusive argument.
Statements of the form "A implies B" are equivalent to Sigma_1 sentences:
"there is a number n with a certain decidable property (n codes a derivation
of B from A)". Even the simplest systems of arithmetic, such as Q, do in
fact prove all true Sigma_1 sentences. But the proofs can be huge ...
For example, instances of Goodstein's Theorem (a Pi_2 sentence of the form
"forall x, exists y R(x,y)") are Sigma_1 sentences, and thus are all
provable in Q. I.e., Q proves G(1), G(2), G(3), etc., But their proofs in Q
(or in PA) would be fantastically huge.
So, if, like any reasonable mathematician, you think that G(100) is true, it
cannot be because you have proved it in PA (despite the fact that G(100) is
provable in PA). It is presumably because you accept the usual proof of the
general claim "for all n, G(n)", using set theory given by Goodstein in
1944, and then you took the instance for 100.

--- Jeff


0
ketland (18)
1/14/2005 4:53:37 AM

tchow@lsa.umich.edu wrote:

>In article <TaHFd.70924$8l.16323@pd7tw1no>,
>namducnguyen  <namducnguyen@shaw.ca> wrote:
>  
>
>>I wonder, though, what would happen if P is a proposition in theory T 
>>that we can't never
>>know if there is a proof in T (notwithstanding that it must either have 
>>or have not
>>a proof in T)? Could "proposition P has a proof in theory T" be a 
>>candidate for an
>>_eternal_ [mathematical] truth?
>>    
>>
>
>If it is permanently unknowable whether P is provable in T, then `P is
>unprovable in T' would be a permanently unknowable truth, yes.  But are
>there plausible candidates for such a P and T?
>  
>

It all depends on what we mean by "plausible" and "candidate". If by
"plausible" we only mean "good suspicion", and by "candidate" we mean
to spell out completely symbol by symbol, then GC - GoldBach's Conjecture -
in PA looks like a candidate.

If, however, by "candidate" we only care for the _existence_ of a formula f
of the L(T) then I think we could do the following. Instead of (Godel's)
numerization of  L(T), we could use (ZFC) "set-ization" - encoding L(T)
in the standard model of ZFC - and use AC to *choose * a formula f which
we know exists [due to AC] but we'd know nothing of its content, hence it
would be impossible to know whether or not f is provable in T.

Those are the only 2 "candidates" I could think of. But my point here is 
that I
don't believe anything of the nature of being "absolute" or "eternal" in
mathematics. Imho, the name "Godel" should remind us when we used to think
that it were absolutely true that a formula must be either have proof,
or disproof. I also think that there is no such a thing as a truth value
standing on its own: "truth" and "falsehood" must always be a function of
interpretation, of assumptions, and the like... whether we're in 
first-order
level, or in meta-level 1, or 2, etc...

It seems that typically, we'd not need think about those too much.
But imho, if we keep using the words "know", "unknownable",
"mathematical truth", etc.. with an implied qualifier "permanent", 
"eternal",
or the like, ...then eventually I'd think our reasoning would run into 
trouble.
0
1/14/2005 5:57:24 AM
namducnguyen wrote in message <8bJFd.72105$8l.14155@pd7tw1no>...

>Imho, the name "Godel" should remind us when we used to think
>that it were absolutely true ....

Reminds me of what, I assume, Lady Bracknell used to wear.

--- Jeff


0
ketland (18)
1/14/2005 6:38:12 AM
tchow@lsa.umich.edu wrote:
> In article <1105491750.589150.193360@c13g2000cwb.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> >This seems silly to me.  Suppose someone asks, "Are the group axioms
> >true?"  Obviously, they are true of any set of objects which satisfy
> >them.  But suppose he asks again, emphatically, "No, but are they
> >*true*?"  What is that to mean?
>
> First of all, if truth doesn't mean anything, then there are
certainly no
> truths at all, let alone permanently unknowable ones.

I didn't go *that* far.   :-)
>
> But more to the point, group axioms aren't the right example to
choose
> here.  I might ask, is it true that every differentiable function is
> continuous?  Is it true that the square root of 2 is irrational?  We
> all say "yes" in ordinary mathematical practice.  That's because we
> can prove them.  But proof is only useful in establishing truth if
the
> axioms that it starts with are true.  Whatever axioms you choose to
> start with (ZFC, 2nd order arithmetic, whatever), the question arises
> as to whether those axioms are true.  If you say that asking about
the
> truth of the axioms is meaningless, then presumably asking about the
> truth of the theorems proved from them is also meaningless, and we're
> back in the situation where there are no truths at all.

(I promise I'll respond to your points)  I think my choice of the group
axioms is a good one because it *appears* that there is a relevant
distinction between the group axioms and ZFC, but there really isn't.
We can easily formalize the axioms of group theory in FOL so that any
model of the axioms is a group (Let's assume, for a moment, that our
model theory is going on in a set theory other than ZF just to avoid
confusion):

1.  ExAy ((x*y = y) & (y*x = y)) (usually denoted 1)
2.  AxEy (x*y = 1)
3.  AxAyAz (((x*y)*z) = (x*(y*z)))

And we can trivially ( *smile* ) axiomatize ZF.  Now, we can say that
ZF is true of a collection of sets iff the collection of sets satisfies
ZF.   We say that the group axioms are true of a set of objects iff
that set of objects satisfies the axioms of group theory.  Logically
(with regards to model theory), nothing different is going on here.
What *may* be different is that we might have an intended structure (in
the non-technical sense) whose properties we intend to capture.  But my
intended structure might be different from yours.  Presumably, neither
should affect the truth of ZF.

But you equivocate this notion of truth with some other when you claim
that ZF must be true simpliciter for a proof to establish truth the
truth.  By the Soundness theorem, given a set of sentences L, there
exists a proof of S from premises in L only if S is true in every model
in which each of the elements of L are true.  Again, this is truth
relative to a model, not truth simpliciter.  However, a proof does
alone does not establish the truth of S.  It establishes the truth of L
-> S.  The truth of S is established by examining its truth value in a
model which satisfies L.

>
> >> The axiom of choice is now accepted as true by most
mathematicians.
> >
> >I don't think "as true" is necessary here.  In fact, I think it
> >obfuscates the issue.  The acceptance of AC is as much a consequence
of
> >sociology as it is of logical consistency.
>
> But in the context of the question of the existence of permanently
> unknowable truths, is AC a known truth?  Is it true that every vector
> space has a basis?  If these things aren't "true," then what is?

That AC implies that every vector space has a basis.  And that if every
vector space has a basis, AC follows.

This is short, I realize.  But perhaps it can clearly communicate what
I meant to above.

'cid 'ooh

0
poopdeville (133)
1/14/2005 7:23:13 AM
poopdeville@gmail.com writes:

> (I promise I'll respond to your points)  I think my choice of the group
> axioms is a good one because it *appears* that there is a relevant
> distinction between the group axioms and ZFC, but there really
> isn't.

  Relevant to what?
0
torkel (478)
1/14/2005 7:26:27 AM
LordBeotian wrote:
> <poopdeville@gmail.com> ha scritto
>
> > > which *axioms* are true.
> > >
> >
> > This seems silly to me.  Suppose someone asks, "Are the group
axioms
> > true?"  Obviously, they are true of any set of objects which
satisfy
> > them.  But suppose he asks again, emphatically, "No, but are they
> > *true*?"  What is that to mean?
>
> When we ask if a (formal) statement is true, we mean "true in a
specific
> model" that we may not specify if it is clear which the model is. If
the
> statements are the group axioms it is unclear what is the model, not
if the
> statement is the axiom of choice.

Of course.  That's what I was trying to get at with my example.
However, there are *many* models of ZFC, and as far as I know, none of
them is particularly special in the sense that it is clear which one we
are talking about when we talk about truth.

'cid 'ooh

0
poopdeville (133)
1/14/2005 7:30:30 AM
tchow@lsa.umich.edu wrote:
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> 
>>when mathematicians make unqualified statements, with no 
 >>other context, they usually *mean* "In ZFC ...".
> 
> As a matter of sociological fact, this is definitely false.  For a trivial
> counterexample, consider:
> 
>   (*)  "sqrt(2) is irrational" is provable in ZFC.
> 
> When mathematicians assert (*), they mean (*).  They don't mean
> 
>   (**) `"sqrt(2) is irrational" is provable in ZFC' is provable in ZFC.
> 
> They couldn't, obviously, because this would instantly lead to an infinite
> regress.

Somehow, this seems unfair. How do we know "sqrt(2) is irrational" is
true? By a proof in, say, ZFC. You might then wonder how we know that
-that- (*) is true, (leading to the regress) but that is not the same
as the knowing the basic statement is true.

> The familiar proof that sqrt(2) is irrational consists of a sequence of
> meaningful statements that we can read and understand and that leads us
> to accept the truth of "sqrt(2) is irrational"; mimicking this proof
> formally in this or that formal system does not yield the "true meaning"
> of the statements in question.

But isn't the proof integral to the meaning? The "true meaning" of a
well-formed, true or false mathematical statement must start
somewhere, and a proof seems to be the most objective source.

> In fact, it's almost the other way around;
> we only accept (as a candidate for foundations) formal systems that
> faithfully mimic what we *already* recognize to be correct reasoning.
> Where do you think ZFC came from in the first place?

OK, so "correct" reasoning came first before the specific 
formalization (or really hypothetical formalization, unless Mizar has 
a proof of the irrationality of sqrt(2)). But the meaning came from 
some sort of reasoning (in the direction of being formal).

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/14/2005 9:08:56 AM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> But isn't the proof integral to the meaning? The "true meaning" of a
> well-formed, true or false mathematical statement must start
> somewhere, and a proof seems to be the most objective source.

  So what is the meaning of Goldbach's conjecture?
0
torkel (478)
1/14/2005 9:14:27 AM
Torkel Franzen wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:
> 
> 
>>But isn't the proof integral to the meaning? The "true meaning" of a
>>well-formed, true or false mathematical statement must start
>>somewhere, and a proof seems to be the most objective source.
> 
> 
>   So what is the meaning of Goldbach's conjecture?

Yeah, yeah, I know. It has some meaning to it.

But we don't have a handle on its -true- meaning (yes, we're playing
with pretty slippery stuff). And often once something is finally
proven (one way or the other), it is often noted that we don't have
the 'true' meaning until pieces of the proof (a proof?) have been
picked apart, or generalized or totally redone in another manner, or
explained to a six-year old, or whatever. There's all sorts of stuff
to do with meaning, but I'm just saying that if you ignore proof,
then you're missing something that is more important than anything
else.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/14/2005 1:50:48 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> But we don't have a handle on its -true- meaning (yes, we're playing
> with pretty slippery stuff).

  "Slippery" isn't the word for it. Rather, your comments make no
apparent sense at all as long as you haven't explained what you
require of "meaning" or of "true meaning".
0
torkel (478)
1/14/2005 1:52:50 PM
Torkel Franzen wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:
> 
>>But we don't have a handle on its -true- meaning (yes, we're playing
>>with pretty slippery stuff).
> 
>   "Slippery" isn't the word for it. Rather, your comments make no
> apparent sense at all as long as you haven't explained what you
> require of "meaning" or of "true meaning".

I was attempting to use those terms consistent with what I was
responding to. So, your interpretation of "meaning" for what they
said did not (also?) make sense for my response? Maybe you can help
me; what did "meaning" or "true meaning" ..ahem.. mean for them?

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/14/2005 3:30:50 PM
Torkel Franzen <torkel@sm.luth.se> writes:

> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:
>
>> But isn't the proof integral to the meaning? The "true meaning" of a
>> well-formed, true or false mathematical statement must start
>> somewhere, and a proof seems to be the most objective source.
>
>   So what is the meaning of Goldbach's conjecture?

Different from what it would be if we had a proof?

" One would like to say: the proof changes the grammar of
  our language, changes our concepts.  It makes new connexions,
  and it creates the concept of these connections. (It does not establish
  that they are there; they do not exist until it makes them.) "

(Wittgenstein, remarks on the foundations of mathematics, III, 31)



-- 
Alan Smaill

0
smaill (3)
1/14/2005 3:51:24 PM
Alan Smaill <smaill@inf.ed.ac.uk> writes:


> Different from what it would be if we had a proof?

  What would be different?
0
torkel (478)
1/14/2005 4:42:53 PM
tchow@lsa.umich.edu wrote:
> In article <cs6ep3$lso$1@ra.nrl.navy.mil>, Ralph Hartley
> <hartley@aic.nrl.navy.mil> wrote:
> 
>> But you are presumably a mathematician, and when mathematicians make 
>> unqualified statements, with no other context, they usually *mean* "In
>> ZFC ...".
> 
> As a matter of sociological fact, this is definitely false. For a
> trivial counterexample, consider:
> 
> (*)  "sqrt(2) is irrational" is provable in ZFC.
> 
> When mathematicians assert (*), they mean (*).  They don't mean
> 
> (**) `"sqrt(2) is irrational" is provable in ZFC' is provable in ZFC.
> 
> They couldn't, obviously, because this would instantly lead to an
> infinite regress.

I did say "usually".

The most obvious exception is most statements about ZFC.

But (*) doesn't need that exception because it is not an "unqualified"
statement.

I could argue that when they say (*) they really mean

(***) `"sqrt(2) is irrational" is provable in ZFC' is provable in V.

Where V is the much weaker system used to verify proofs.

Are there any cases where you need the axiom of choice to prove that
something is provable in ZFC? (That's different from a proof that uses AC)
It would be odd, the normal way to prove something is a theorem is to
exhibit a proof, and checking a proof is a finite thing.

Only in such a rare case of a non-constructive proof of theoremhood would
someone say (or mean) "'A is provable in ZFC' is provable in ZFC". I have 
seen statements of the form '"A is not provable in ZFC" is not provable in 
ZFC'.

I *could* argue that, but I don't think I will, because then the question
of in what theory (***) is provable might come up.

You are the the final arbiter of what you mean by a statement, but when *I*
say something like "sqrt(2) is irrational", I think I might mean something
like:

I^1 have seen and verified a proof^2 in ZFC^3 of "sqrt(2) is irrational".

1) Someone I *really* trust has, in this case I have myself, but I don't
even trust myself completely.

2) Actually a summary. I don't think I have seen (much less verified) a
proof of every single step from the axioms of ZFC, but I am confident that
there is one, and that I could produce one if I really needed one. In this 
case I think I even know where to find one.

3) I only mean "ZFC" by default. The proof may not have used every axiom of
ZFC (in this case I know it didn't) so it is really a proof in a weaker
system. I am (intentionally or not) making a slightly ambiguous statement,
because it is impractical to always specify exactly what my conclusions
depend on.

If pressed, I would surely have to add even more footnotes. Most of the
time it is enough to say (and think) "sqrt(2) is irrational."

> Even you seem comfortable with asserting (*) flat out; that is, it isn't
> a white lie or a shorthand for something like (**)---it is meaningful on
> its own.

Maybe it's a *little* bit of a white lie, since I *ought* to note that it
really depends on V, and I don't. But V is so minimal that I feel justified
ignoring it completely. If you start with nothing, and trust no
assumptions, it is impossible to think at all.

> So we have at least one class of examples of mathematical statements
> that are meaningful on their own.  Mathematicians typically include
> "sqrt(2) is irrational" and "every differentiable function is
> continuous" and so forth among the mathematical statements that are 
> meaningful on their own, and that are *true* in an absolute sense, just 
> as (*) is true in an absolute sense.

Just as they once included the Parallel Postulate.

(*) is dependent on fewer assumptions than "sqrt(2) is irrational", so one
might say that (*) is *more* absolutely true, if one could do so without
cruelly abusing words.

Except on Usenet :-), I could convince a finitist  of (*), but not of
"sqrt(2) is irrational".

> There's another way to see that ZFC doesn't, in practice, have the
> status that you assign it.  Suppose that someone were to find a
> contradiction in ZFC.  Would this make any difference to mathematics?
> It would depend on the specific contradiction, but in general, it
> wouldn't make any difference.

I think it would make a huge difference.

I think there are proofs (not in ZFC) that ZFC is consistent. I'm not sure
exactly what those proofs depend on, but if ZFC is inconsistent, then those
proofs must not be sound either.

You couldn't just give up choice or whatever,  and continue like nothing
had happened.

For the rest of this post I will assume not that ZFC had been found to be 
inconsistent, but instead that mathematicians decided that ZFC does not 
capture their intuitive idea of "sets" very well after all, and *decided* 
to use something else. I think your arguments apply just as well to that 
example, and it could happen.

Someone would have to go back through every proof and make sure it
is still valid (or a valid proof exists). Many would be easy, and some 
could be handled in bulk. For some axioms (e.g. choice) it may be customary 
to keep track, which would make it easier.

> Logicians would just pick some other foundation for mathematics with no
> known contradiction.  ZFC is way too strong for most of ordinary
> mathematics anyway.  All the theorems in the books would remain intact,
> except for the few that were affected by the specific contradiction.

It would make a *big* difference where the problem was. Just giving up
choice would be easy.

But suppose the problem were in V.

That would infect *all* proofs, and mathematics would have to start over. 
(It is unlikely they would do that voluntarily.)

> What would you say then?  That in this new situation, "sqrt(2) is
> irrational" no longer means "`sqrt(2) is irrational' is provable in ZFC"
> but now means "`sqrt(2) is irrational' is provable in X," where X is the
> new foundation? Given that the usual proof of sqrt(2)'s irrationality is
> left unchanged by the discovered contradiction in ZFC, it's a little
> bizarre to think that its meaning has changed.

My footnote 3) above supplies some wiggle room. The meaning of an ambiguous
statement need not be perfectly fixed.

> ... mimicking this proof formally in this or that formal system does not
> yield the "true meaning" of the statements in question.
> In fact, it's almost the other way around; we only accept (as a
> candidate for foundations) formal systems that faithfully mimic what we
> *already* recognize to be correct reasoning. Where do you think ZFC came
> from in the first place?

If you look at axioms as definitions, that makes sense. A formal definition 
should match the informal definition as well as is practical, or it is no good.

The formal system does not yield "the true meaning", it yields "an 
unambiguous meaning" (or less ambiguous).

When someone says "sqrt(2) is irrational" it may be a bit ambiguous. Is it 
a statement about their intuitive (but imprecise) idea of the reals, or 
about the much more exactly defined objects of a formal system?

If the definitions provided by the formal system are good enough, those two 
meanings (mostly) coincide, and it doesn't matter (much).

Ralph Hartley
0
hartley (156)
1/14/2005 5:16:11 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> I was attempting to use those terms consistent with what I was
> responding to. So, your interpretation of "meaning" for what they
> said did not (also?) make sense for my response? Maybe you can help
> me; what did "meaning" or "true meaning" ..ahem.. mean for them?

  Well, it would be preferable if you yourself meant something by
"true meaning".
0
torkel (478)
1/14/2005 5:32:50 PM
Torkel Franzen <torkel@sm.luth.se> writes:

> Alan Smaill <smaill@inf.ed.ac.uk> writes:
>
>
>> Different from what it would be if we had a proof?
>
>   What would be different?


It would play a different role in the mathematical language game.

If we take W's position on meaning to heart, then it follows
that the proved statement means something different from the unproved one.

I know, I know, it's an extreme position ....

-- 
Alan Smaill    
School of Informatics             tel:   44-131-650-2710
University of Edinburgh           
0
smaill1 (89)
1/14/2005 5:34:19 PM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> Are there any cases where you need the axiom of choice to prove that
> something is provable in ZFC?

  No.

> When someone says "sqrt(2) is irrational" it may be a bit ambiguous. 

  Any statement "may" be ambiguous.
0
torkel (478)
1/14/2005 5:36:02 PM
Alan Smaill <smaill@SPAMinf.ed.ac.uk> writes:


> It would play a different role in the mathematical language game.

  This bit of jargon means nothing in particular.
0
torkel (478)
1/14/2005 5:36:26 PM
Torkel Franzen <torkel@sm.luth.se> writes:

> Alan Smaill <smaill@SPAMinf.ed.ac.uk> writes:
>
>
>> It would play a different role in the mathematical language game.
>
>   This bit of jargon means nothing in particular.

Well, it's not a mathematically defined notion, I agree.

But I do think that the notion serves as motivation for more
systematic work on the theory of meaning, without depending
on it at the end of the day.  I'm thinking of Dummett's
work in particular.

Whether any form of this takes us to a place where meaning
changes on finding proof is of course a different question.

-- 
Alan Smaill 

0
smaill1 (89)
1/14/2005 6:18:37 PM
Alan Smaill <smaill@SPAMinf.ed.ac.uk> writes:

> But I do think that the notion serves as motivation for more
> systematic work on the theory of meaning, without depending
> on it at the end of the day.  I'm thinking of Dummett's
> work in particular.

  Dummett's ideas about a theory of meaning do not imply that the
meaning of Goldbach's conjecture changes when it has been proved.
Rather, the meaning of Goldbach's conjecture, on such a view, is
determined by how it can be proved.


0
torkel (478)
1/14/2005 6:35:03 PM
In article <cs8ug1$mu7$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> I^1 have seen and verified a proof^2 in ZFC^3 of "sqrt(2) is
> irrational".  [...] I don't think I have seen (much less verified) a
> proof of every single step from the axioms of ZFC, but I am confident
> that there is one, and that I could produce one if I really needed
> one. In this case I think I even know where to find one.

Indeed. Freek Wiedijk's _ The Sixteen Provers of the World _
<http://www.cs.ru.nl/~freek/comparison/> presents 16 computer-verified
formal proofs of the irrationality of sqrt(2).

I believe only the Metamath proof shows "every single step".
-- 
Josh Purinton
0
1/15/2005 12:06:20 AM
Torkel Franzen wrote:
> poopdeville@gmail.com writes:
>
> > (I promise I'll respond to your points)  I think my choice of the
group
> > axioms is a good one because it *appears* that there is a relevant
> > distinction between the group axioms and ZFC, but there really
> > isn't.
>
>   Relevant to what?

Relevant to the existence of truth vis a vis intended models versus
plain-old models.

'cid 'ooh

0
poopdeville (133)
1/15/2005 12:09:22 AM
poopdeville@gmail.com writes:

> Relevant to the existence of truth vis a vis intended models versus
> plain-old models.

  That's a bit cryptic, and insufficient to explain such a startling
statement. Since models of ZFC is a very specialized topic, which
is irrelevant to the work of most mathematicians, whereas every
mathematician is familiar with groups, saying that there is
"no relevant distinction" between the two subjects calls for a bit
of elaboration.





  
0
torkel (478)
1/15/2005 7:23:49 AM
Torkel Franzen wrote:
> poopdeville@gmail.com writes:
>
> > Relevant to the existence of truth vis a vis intended models versus
> > plain-old models.
>
>   That's a bit cryptic, and insufficient to explain such a startling
> statement. Since models of ZFC is a very specialized topic, which
> is irrelevant to the work of most mathematicians, whereas every
> mathematician is familiar with groups, saying that there is
> "no relevant distinction" between the two subjects calls for a bit
> of elaboration.

Fair enough.  I'll see if I can.  This is a bit hazy to me as well.
:-)

There are two fairly obvious distinctions between the two, other than
the fact that the axioms of ZF and of group theory are different in
content.  The first is the one you noted: as a matter of mathematical
practice, mathematicians are more familiar with models of groups than
with models of ZF.  The second is more linguistic in nature.  A
mathematician asked if the axioms of group theory are true would likely
note, as we have all noted, some awkwardness in the way the question
was phrased.  This is related to the first distinction in that even if
the mathematician isn't thinking about interpretations and structures
and all that jazz, he is thinking about the axioms being true *of
something;* namely, particular groups.  (Which of course are models in
the logical sense)

Paraphrasing,  amongst other things, Tim Chow asked if AC and the other
axioms of ZF are true.  Unless he was using a non-model-theoretic use
of the term "true," the ZFC is just as true as the group axioms, since
we can exhibit models for both sets of axioms.  Via forcing, we can
construct a model where ZF holds but AC fails, and similarly, we can
construct a model where only two out of the three group axioms hold.
Neither of the distinctions is relevant to the existence of such
models.

If Tim wasn't using the model-theoretic notion of truth above, then he
wasn't particularly clear what he meant.  One reasonable disambiguation
is whether or not the "intended model" of set theory satisfies the
axioms -- whether or not ZF satisfies our intuitive notions of what it
means to be a set.  As evidenced by the plethora of intuitive notions
of sets on sci.logic, there is no unique intended model.  This relates
to the point I made regarding the acceptance of AC among mathematicians
-- although there are many intended models, there is enough overlap
across intended models of set theory (either by natural intuition,
education, indoctrination, or some other sociological phenomenon) so
that AC is accepted by a majority of, but not all, mathematicians.

Another reasonable disambiguation is that Tim was referring to truth
simpliciter -- a phrase that I've heard others use and describe, but
that I find meaningless.

'cid 'ooh

0
poopdeville (133)
1/15/2005 9:07:07 AM
poopdeville@gmail.com writes:

> Paraphrasing,  amongst other things, Tim Chow asked if AC and the other
> axioms of ZF are true.  Unless he was using a non-model-theoretic use
> of the term "true," the ZFC is just as true as the group axioms, since
> we can exhibit models for both sets of axioms.

  If by a "model-theoretic use of 'true'" you mean a use whereby the
axioms of a theory are called "true" if they have a model, there is
no such usage in logic or mathematics. So obviously Tim was using a
"non-model-theoretic" sense of "true". You are reluctant to speak of
the axioms of ZFC as true (except in your Pickwickian sense), but why
is this? Would you say that since we can exhibit a model of
PA+"PA is inconsistent", the axioms of this theory are "just as true
as the group axioms"?
0
torkel (478)
1/15/2005 9:54:25 AM
And what is the sense of "true" that is used in mathematics? Maybe you
would like to enlighten us, and end a long running philosophical debate
in the process.

--
Eray

0
examachine (384)
1/16/2005 10:31:38 AM
<poopdeville@gmail.com> ha scritto

> > When we ask if a (formal) statement is true, we mean "true in a
> specific
> > model" that we may not specify if it is clear which the model is. If
> the
> > statements are the group axioms it is unclear what is the model, not
> if the
> > statement is the axiom of choice.
>
> Of course.  That's what I was trying to get at with my example.
> However, there are *many* models of ZFC, and as far as I know, none of
> them is particularly special in the sense that it is clear which one we
> are talking about when we talk about truth.

I agree when you say that it is not very clear what the "standard" model of
set theory is made of (in fact it is an open problem if CH has to be
accepted or not). But it's not true that none of the models is particulary
special: the univers of costructible set for example looks much more
interesting (for a mathematician) than (for example) the model where you are
the empty set and I am omega...





0
1/16/2005 10:44:18 AM
"namducnguyen" <namducnguyen@shaw.ca> ha scritto

> If, however, by "candidate" we only care for the _existence_ of a formula
f
> of the L(T) then I think we could do the following. Instead of (Godel's)
> numerization of  L(T), we could use (ZFC) "set-ization" - encoding L(T)
> in the standard model of ZFC - and use AC to *choose * a formula f which
> we know exists [due to AC] but we'd know nothing of its content, hence it
> would be impossible to know whether or not f is provable in T.

That looks intesting (really)... I would like to understand how set-ization
is made and how AC is needed to choose  a single formula (typically we use
AC to make infinite choices).




0
1/16/2005 10:47:22 AM
Torkel Franzen  <torkel@sm.luth.se> wrote:
>Alan Smaill <smaill@SPAMinf.ed.ac.uk> writes:
>
>> But I do think that the notion serves as motivation for more
>> systematic work on the theory of meaning, without depending
>> on it at the end of the day.  I'm thinking of Dummett's
>> work in particular.
>
>  Dummett's ideas about a theory of meaning do not imply that the
>meaning of Goldbach's conjecture changes when it has been proved.
>Rather, the meaning of Goldbach's conjecture, on such a view, is
>determined by how it can be proved.

Without any justification, I think the meaning of a statement changes in 
the process of going from conjecture to theorem or nontheorem, whatever 
Dummett's implications are (I am not familiar with them). And this does 
not seem inconsistent with your second claim about Dummett (though I find 
it hard to pin down).

Mitch


0
harrisq (267)
1/16/2005 2:12:43 PM
harrisq@tcs.inf.tu-dresden.de (Mitch Harris) writes:

> Without any justification, I think the meaning of a statement changes in 
> the process of going from conjecture to theorem or nontheorem, whatever 
> Dummett's implications are (I am not familiar with them).

  Well, whatever doctrine about meaning you have in mind here, it is
not that of Dummett.


0
torkel (478)
1/16/2005 2:29:06 PM
examachine@gmail.com wrote in message
<1105871498.331035.215870@z14g2000cwz.googlegroups.com>...
>And what is the sense of "true" that is used in mathematics? Maybe you
>would like to enlighten us, and end a long running philosophical debate
>in the process.

Alfred Tarski enlightened us on these matters, over 70 years ago, with an
important work called "Der Wahrheitsbegriff in den formalisierten Sprachen".
In the process he managed to clarify and introduce many deep ideas and prove
important theorems about the foundations of semantics and the notion of
truth (e.g., the criterion of material adequacy, the structure of truth
definitions, the indefinability of arithmetic truth in the first-order
language of arithmetic, the possibility of constructing truth definitions in
richer languages---e.g., higher-order or by infinitary devices). This work
has been extended in various ways (e.g., by work on self-applicative truth
theories by Kripke and others).

This is all well-known, accepted by everybody with at least a minimal grasp
of mathematical logic (as evinced round here (sci.logic) by Daryl, Tim,
Torkel and Aatu) and is what anyone means when they ask, for example, "Is GC
true?". For, GC is true iff, for any even number n>2, there are primes p, q
such that n = p + q. Do you think that every even number >2 might have this
property and GC still be untrue? Do you think that GC could be true even
though there is a counter-example? In other words, do you disagree with the
T-sentence

    GC is true iff, for any even number n>2, there are primes p, q such that
n = p + q

(The question of whether GC can be proved is quite separate.)

Perhaps you can "enlighten" us with your own criticisms of this standard and
well-known work in semantic theory. For example: suppose you think
arithmetic truth is extensionally equivalent to some property X (which you
have an obligation to explain).
Then: how would you verify the T-scheme for this property X? I.e., how would
you prove, for each arithmetic sentence A, every instance of

  (T)  "A" has property X if and only if A

(where "A" is a structural descriptive name of the sentence A).

Incidentally, there *do* exist important criticisms of this standard work in
semantical theory (e.g., by Dummett and his school; and by Putnam; and
others). But, you appear not to have the foggiest acquaintance with even the
most elementary issues here, and show no evidence of having read anything on
the matter.

I suggest that you go and buy any standard book on these matters. For
example, Richard Kirkham's "Theories of Truth" (1992, MIT Press). You might
look at some of the recent anthologies (e.g., the one edited by Simon
Blackburn and Keith Simmons, or the admirable recent anthology edited by
Michael Lynch).
If you would like to obtain a better understanding of semantic theory, there
is nothing stopping you from going and reading this work. Had Karl Marx
studied semantic theory, he might have put it as follows: You have nothing
to lose but your (mental) chains.

--- Jeff


0
ketland (18)
1/16/2005 8:46:29 PM
In article <34vb2rF4c9st7U1@news.dfncis.de>,
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote:
>Without any justification, I think the meaning of a statement changes in 
>the process of going from conjecture to theorem or nontheorem

Since you're explicitly disavowing any attempt at justification, perhaps I'm
being mean to attack your statement, but I must say that this goes very
strongly against common sense.  Common sense would suggest that the meaning
of a statement, mathematical or otherwise, does not depend on whether we
have good justification for believing it.  Someone tells me, "The fastest
way to get to the museum is to take a right at the next light."  This is
meaningful; knowing English, I understand what is being claimed.  The
statement might be true or false and I may or may not have adequate
justification for believing it, but it is very strange to say that the
*meaning* of the statement depends on this.  Suppose I don't know whether
to trust this person; does that mean that the directions are *ambiguous*,
and that if the person turns out to be trustworthy then the directions
will mean one thing, and if not then the directions will mean something
else?  Very strange.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/17/2005 3:06:26 AM
In article <34pgh8F4dp4aoU1@news.dfncis.de>,
Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>Somehow, this seems unfair. How do we know "sqrt(2) is irrational" is
>true? By a proof in, say, ZFC. You might then wonder how we know that
>-that- (*) is true, (leading to the regress) but that is not the same
>as the knowing the basic statement is true.

I was talking about what "sqrt(2) is irrational" *means*, which is quite
different from asking whether it is *true*.  See elsewhere for my response
to your attempt to say that meaning depends on whether or not something has
been proved.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/17/2005 3:09:22 AM
In article <cs8ug1$mu7$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>tchow@lsa.umich.edu wrote:
[Re: claim that mathematicians usually mean "X is provable" when they say "X"]
>> As a matter of sociological fact, this is definitely false.
[...]
>I did say "usually".

Again, as a matter of sociological fact, this is just totally false.  *You*
might usually mean "X is provable" when you say "X," but your claim was
about mathematicians, and the vast majority of mathematicians do not share
your point of view.  You might be right and they might be wrong, but I am
only pointing out the sociological fact about mathematicians.

>I think there are proofs (not in ZFC) that ZFC is consistent. I'm not sure
>exactly what those proofs depend on, but if ZFC is inconsistent, then those
>proofs must not be sound either.

You can prove, for example, in "ZFC + there exists a strongly inaccessible
cardinal" that ZFC is consistent.  So of course, if you give up ZFC, then
you would have to give up "ZFC + some other axioms."  This would not make
much of a difference to most of mathematics, because very little mathematics
makes use of those other axioms.  As I said, most of mathematics doesn't use
anywhere near the full power of ZFC.  First-order Peano arithmetic, for
example, is good enough for a huge majority of mathematics.

[Re: If ZFC is abandoned]
>Someone would have to go back through every proof and make sure it
>is still valid (or a valid proof exists).

Nope.  You make it sound like the way a mathematician checks a proof is to
verify that it is provable in ZFC.  This never happens outside of logic and
automated theorem proving, and even in logic papers it rarely happens.
Mathematicians just check proofs by reading them and convincing themselves
of their logical correctness.  They do not verify their provability within
any specific formal system, unless they are actually verifying "X is
provable in ZFC" as opposed to verifying X itself.

Since they never did what you seem to think they did in the first place,
they certainly aren't going to go back and do it over.

>It would make a *big* difference where the problem was. Just giving up
>choice would be easy.

Yes, it would.  If the contradiction were in a very weak system, then that
would make a difference.  But anyone other than someone with an extremely
skeptical philosophical stance knows with certainty that this can't happen.
There is hardly anything in the world that we know with greater certainty
than things we have proved mathematically.

>When someone says "sqrt(2) is irrational" it may be a bit ambiguous. Is it 
>a statement about their intuitive (but imprecise) idea of the reals, or 
>about the much more exactly defined objects of a formal system?

It need not be a statement about the reals; indeed, its most natural
reading is equivalent to, "Given any strictly positive integers a and b,
the integers a^2 and 2b^2 are distinct."  This statement is about as precise
a statement as there is in this world.  If you regard this statement as
imprecise, then the definition of a "formal system" is also imprecise,
since formal systems are defined in terms of things like "symbols," "rules,"
"strings," and so forth, which are no more precise than "integers."
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/17/2005 3:29:32 AM
In article <1105780027.473885.47200@c13g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Paraphrasing,  amongst other things, Tim Chow asked if AC and the other
>axioms of ZF are true.  Unless he was using a non-model-theoretic use
>of the term "true," the ZFC is just as true as the group axioms, since
>we can exhibit models for both sets of axioms.

I used the term the way mathematicians normally use them, which never causes
problems except when someone suddenly decides to get skeptical and ask,
following Pontius Pilate, "What is truth?"  For the present purposes, I
can eliminate all uses of the word "true" by simply replacing the statement
of which I am predicating truth with the statements themselves.  This is
cumbersome, but is helpful psychologically for those who aren't practiced
in doing such things themselves.  So when I ask if AC is true, I am asking
if the Cartesian product of nonempty sets is nonempty.  When I ask if "ZFC
is consistent" is true, I am asking if ZFC is consistent.  When I ask if
"`Every vector space has a basis' is provable in ZFC" is true, I am asking
if `Every vector space has a basis' is provable in ZFC.

So if you claim not to understand what I mean when I say that "ZFC is
consistent" is true, even after you understand how to eliminate the word
"true" as I have just demonstrated, then you are really claiming not
to understand what I mean when I say that ZFC is consistent.  And of
course, in this example, you *do* understand what I mean, because you
make similar assertions yourself, like "ZFC is consistent if and only if
it has models," which you presumably wouldn't make if you didn't know
what such an assertion meant, and which presupposes that you know what
it means for ZFC to be consistent.

The difference between ZFC and the axioms for group theory is not any
kind of interesting structural difference between the first-order theories
themselves, as you point out.  The difference is that ZFC is usually used
in the context of trying to capture certain features of general mathematical
discourse---in particular, statements that we make all the time that we feel
we understand the meaning of unambiguously.  Although maybe you have not
articulated it to yourself explicitly, I bet you in fact believe that you're
making a specific, meaningful statement when you say "`Every vector space
has a basis' is provable in ZFC."  You do not think, "Gosh, what do I really
mean by such a statement?  Do I mean that the statement `"Every vector space
has a basis" is provable in ZFC' is true in some model of some formal
system?  Which model of which formal system am I talking about?"

The group-theoretic axioms, however, are not introduced in order to capture
statements that mathematicians assert directly without any sense of
ambiguity.  Mathematicians do not go around saying things like, "For every
x, y, and z, (x * y) * z = x * (y * z)" without further explanation about
what x, y, and z are and what * is.

The fact that I can take two different "naturally occurring" phenomena
---general mathematical discourse (statements that mathematicians make with
no sense of ambiguity) and statements that apply in the context of groups---
and use the same tool (first-order logic) to analyze both of them does not
mean that the two naturally occurring phenomena are essentially the same,
any more than the fact that I can look at either the moon or a star through
a telescope makes the moon a star.

For example, you can create models of ZFC in which "ZFC is consistent" is
false.  Does this make you wonder, "What does it *mean* for ZFC to be
consistent?"  (Recall the definitional equivalence between "ZFC is
consistent" and "`ZFC is consistent' is true" that I'm positing.)  It
shouldn't.  You've known all along what you *mean* by "ZFC is consistent."
The fact that someone can mimic your statement in the first-order language
of set theory and then construct models of ZFC in which your statement
holds and models in which it doesn't is no reason to shake your belief
that you know what it means for ZFC to be consistent.

Similarly, mathematicians agree that they know what it *means* for every
vector space to have a basis (even if they don't agree that it's true).
Mimicking this statement in some formal system doesn't automatically make
it ambiguous.  You might, of course, think that the statement really *is*
ambiguous.  But this would be for philosophical reasons and not mathematical
ones, and in particular, independence results do not imply that such
statements are ambiguous.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/17/2005 4:08:19 AM
tchow@lsa.umich.edu wrote:
> Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
> 
>>Somehow, this seems unfair. How do we know "sqrt(2) is irrational" is
>>true? By a proof in, say, ZFC. You might then wonder how we know that
>>-that- (*) is true, (leading to the regress) but that is not the same
>>as the knowing the basic statement is true.
> 
> I was talking about what "sqrt(2) is irrational" *means*, which is quite
> different from asking whether it is *true*.  See elsewhere for my response
> to your attempt to say that meaning depends on whether or not something has
> been proved.

I am not treating "meaning" technically (that is, I personally don't 
know of a technical meaning for it), so my usage of the term can 
easily be problematic (to invoke Wittgenstein again in this context, 
like the term "game").

But whatever my usage is, I do consider that (the technical term) 
truth is part of meaning, that is the meaning of a mathematical 
statement includes its truth value. So "quite different" I don't see 
as "quite apart" but rather "quite more". Is that nonstandard or 
contrary to common sense?

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/17/2005 10:02:14 AM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> But whatever my usage is, I do consider that (the technical term) 
> truth is part of meaning, that is the meaning of a mathematical 
> statement includes its truth value. 

  This is a very weird usage, not in agreement with ordinary usage or
with any theory of meaning that I know of.

0
torkel (478)
1/17/2005 10:03:51 AM
tchow@lsa.umich.edu wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote:
> 
>>Without any justification, I think the meaning of a statement changes in 
>>the process of going from conjecture to theorem or nontheorem
> 
> Since you're explicitly disavowing any attempt at justification, perhaps I'm
> being mean to attack your statement,

Not at all. I was being defensive. Since you did me the favor of
responding I will go into more detail.

> but I must say that this goes very
> strongly against common sense.  Common sense would suggest that the meaning
> of a statement, mathematical or otherwise, does not depend on whether we
> have good justification for believing it.

OK. I realize now that one of the usages of "meaning" is mathematical, 
another is psychological, and, to the extent that they are separable, 
I have been intending the latter.

> Someone tells me, "The fastest
> way to get to the museum is to take a right at the next light."  This is
> meaningful; knowing English, I understand what is being claimed.  The
> statement might be true or false and I may or may not have adequate
> justification for believing it, but it is very strange to say that the
> *meaning* of the statement depends on this.  Suppose I don't know whether
> to trust this person; does that mean that the directions are *ambiguous*,
> and that if the person turns out to be trustworthy then the directions
> will mean one thing, and if not then the directions will mean something
> else?  Very strange.

Not at all strange psychologically, or to force technical meaning on
this, modally.

Back to the sqrt(2) example, what is the mathematical meaning of its
irrationality, -if one erases from our knowledge, its proof and/or
truth value-? We can surely use it hypothetically (it is well-formed).
I feel stuck in the "proof says it all"/"formality" vein, and I am
looking for some meaning outside of that point of view.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/17/2005 10:16:29 AM
Torkel Franzen wrote:
> 
>   G�del regarded it as a kind of miracle that the informal concept of
> mechanical computability could in fact be captured in a formal
> definition. Provability is much more problematic. There was no
> disagreement in mathematics over particular algorithms - it was clear
> to everybody that Sturm's algorithm was an algorithm, that Euclid's
> algorithm was an algorithm, and so on. In the case of provability,
> there is disagreement over what constitutes a proof, and there is
> a distinction between more or less conclusive or convincing proofs.

Do you have any references/can you provide any further explanation
for these? the miracle: was that Goedel's reaction to Turing's
undecidability result? What is the "more or less convincing" you are
referring to; is that the range of formality or something else (is
that a technical disagreement or philosophical one)?

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/17/2005 10:42:50 AM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> Do you have any references/can you provide any further explanation
> for these? the miracle: was that Goedel's reaction to Turing's
> undecidability result?

  No, to the fact that it turned out to be possible to give an absolute
definition of computability. Wang's books should contain the
statements I have in mind.

> What is the "more or less convincing" you are
> referring to; is that the range of formality or something else (is
> that a technical disagreement or philosophical one)?

  I had in mind the principles used in proofs, ranging, say, from
mathematical induction to large cardinal axioms.
0
torkel (478)
1/17/2005 11:03:52 AM
Mitch Harris wrote:

> Do you have any references/can you provide any further explanation
> for these? the miracle: was that Goedel's reaction to Turing's
> undecidability result? 

There's an apparent obstacle to formulating an absolute notion of 
computability. To see this assume we have defined some notion of 
"computability". Arrange then the "computable" functions in a list

  F_1
  F_2
  .
  .
  .
  F_n
  .
  .
  .

And consider the function

  F(x) = F_x(x) + 1

Since all the functions F_n are "computable", it seems F is intuitively 
also computable. But F must differ from every F_n, and hence there is a 
computable function missed by the particular definition. There's a story 
according to which Stephen Kleene become convinced of the correctness of 
Church's thesis after trying to diagonalize out of the class of 
recursive functions and noticing that it can't be done (in the fashion 
outlined above), because the enumeration of total recursive functions is 
not recursive.

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
0
1/17/2005 11:27:08 AM
Torkel Franzen wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:
> 
>>But whatever my usage is, I do consider that (the technical term) 
>>truth is part of meaning, that is the meaning of a mathematical 
>>statement includes its truth value. 
> 
>   This is a very weird usage, not in agreement with ordinary usage or
> with any theory of meaning that I know of.

Then what is the ordinary usage (or what is a description of a 
relevant theory of meaning)?

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/17/2005 12:14:15 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> Then what is the ordinary usage (or what is a description of a 
> relevant theory of meaning)?

  Ordinary talk of "meaning" is extremely varied, but I don't think
you'll find in it any support for the idea that "the meaning of a
mathematical statement includes its truth value". The idea of a
(systematic) "theory of meaning" is chiefly associated with Davidson
and with Dummett. There is a considerable philosophical literature on
the topic, if you're interested.



0
torkel (478)
1/17/2005 2:32:11 PM
Torkel Franzen wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:
> 
>>Then what is the ordinary usage (or what is a description of a 
>>relevant theory of meaning)?
> 
>   Ordinary talk of "meaning" is extremely varied, but I don't think
> you'll find in it any support for the idea that "the meaning of a
> mathematical statement includes its truth value". The idea of a
> (systematic) "theory of meaning" is chiefly associated with Davidson
> and with Dummett. There is a considerable philosophical literature on
> the topic, if you're interested.

This confuses me. From a cursory search of the questionable web, I
found:

   http://www.poetrymagic.co.uk/advanced/david.html

  "Can anything further be done? [beyond Tarski's concept of truth]
The American philosopher Donald Davidson made an enterprising
attempt. His goal is meaning, a clear, unambiguous concept of
meaning, and this he defined (audaciously) as the truth conditions of
a sentence."

Also, further search

   http://www.jtb-forum.pl/jtb/papers/ts_japtom.pdf

seems to say that Dummet objects to Davidson's theory:

"An account of linguistic practice requires the concept of 
recognising-as-true, that of accepting-as-true and that of 
acting-on-the-truth-of. It is unclear that it needs the concept of 
being true (Dummett 1998a, p. 23)."

Whether the author is faithful to Dummett or not, and whether 
Dummett's theory is more justifiable than Davidson's or not, it does
seem to present a situation where -somebody- (not Dummett) supports
the theory that some sort of mathematical truth is involved with some
sort of mathematical meaning.

This seems quite similar to what I was saying (trying to say?). Or am 
I reading (and/or writing) similar words in wholly inappropriate ways 
(and if so, how am I doing that?).

Granted, a web search ain't exactly scholarship ;)
-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/17/2005 3:24:48 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> writes:

> seems to say that Dummet objects to Davidson's theory:

  Yes indeed. These are the two traditional "schools" as regards the
idea of a theory of meaning.

> Whether the author is faithful to Dummett or not, and whether 
> Dummett's theory is more justifiable than Davidson's or not, it does
> seem to present a situation where -somebody- (not Dummett) supports
> the theory that some sort of mathematical truth is involved with some
> sort of mathematical meaning.
> 
> This seems quite similar to what I was saying (trying to say?).

  Not on the face of it. That "truth is involved with meaning" is surely
uncontroversial on any view. Whether or not "Fnork is blorgish" is
true depends, among other things, on what "Fnork" and "blorgish" mean.
0
torkel (478)
1/17/2005 3:31:04 PM
Torkel Franzen <torkel@sm.luth.se> writes:

> Alan Smaill <smaill@SPAMinf.ed.ac.uk> writes:
>
>> But I do think that the notion serves as motivation for more
>> systematic work on the theory of meaning, without depending
>> on it at the end of the day.  I'm thinking of Dummett's
>> work in particular.
>
>   Dummett's ideas about a theory of meaning do not imply that the
> meaning of Goldbach's conjecture changes when it has been proved.
> Rather, the meaning of Goldbach's conjecture, on such a view, is
> determined by how it can be proved.

I agree that the work has no such implication
(and indeed said that there were two questions involved).

-- 
Alan Smaill  

0
smaill1 (89)
1/17/2005 4:10:59 PM

LordBeotian wrote:

>"namducnguyen" <namducnguyen@shaw.ca> ha scritto
>
>  
>
>>If, however, by "candidate" we only care for the _existence_ of a formula
>>    
>>
>f
>  
>
>>of the L(T) then I think we could do the following. Instead of (Godel's)
>>numerization of  L(T), we could use (ZFC) "set-ization" - encoding L(T)
>>in the standard model of ZFC - and use AC to *choose * a formula f which
>>we know exists [due to AC] but we'd know nothing of its content, hence it
>>would be impossible to know whether or not f is provable in T.
>>    
>>
>
>That looks intesting (really)... I would like to understand how set-ization
>is made and how AC is needed to choose  a single formula (typically we use
>AC to make infinite choices).
>  
>

The caveat is that this set-ization is only "syntactical" and doesn't 
involve
any relationship between the hypothesis and conclusion of proof of
a formula f, should f be a theorem. Remember the objective here is to show
the _existence_ of one formula in one theory T that we don't know
"whether or not f is provable in T". The suggested strategy could only work
if the following 2 conditions exist:

(a) T is incomplete. We assume this. [Remember we just need to come up with
    one theory]
(b) for any given theorem p1 [in T], we can always come up with p2, p3
    such that:

    (b.1) all p1,p2,p3 are mutually distinct.
    (b.2) one of p2, p3 is decidable, the other is not.

Conditions (b.1) and (b.2) are necessary to guarantee that we have
an _infinite_ numbers of distinct decidable and undecidable, which is a
necessity for AC to choose a formula that we know that we don't know whether
or not that chosen formula is [un]decidable. But the proof for (b.2) 
exists:
for any given theorem p1, and any undecidable p' [whose existence is
guaranteed by (a)], then by some known rules of inference we have
(p2 = p1 \/ p') be decidable, while (p3 = p1 /\ p') be not.

For the "syntactical" set-ization, or coding the syntax of T using ZFC, 
we just
use a set S that's indexed by the normal Von Neumann's set, to represent
logical and non-logical symbols of T. So the 1st order symbols,
plus - potentially - some non-1st-order ones, are just s1, s2, s3, ....
all of which are in S.  And any (finite) string then would be just
an n-tuple (s1, s2, .., sn) which is a  set. The long and the short of it is
that although each (well-formed)  formula is finite, the [zfc] 
collection/set
F of the formulae is infinite  [and this addresses your concern about the
"infinite choices"], and AC could  now choose one particular f. But by 
(b.1)
and (b.2) we'd have infinite numbers of formulae both ways: decidable and
undecidable. Hence the chosen f could not be known to be in which way.

>
>
>
>  
>
0
1/17/2005 5:37:43 PM
We should all try to be freed of our mental chains indeed. Thanks for
this thoughtful reply. (A longer reply will follow shortly)

--
Eray

0
examachine (384)
1/17/2005 7:58:23 PM
> There's an apparent obstacle to formulating an absolute notion of 
> computability. To see this assume we have defined some notion of 
> "computability". Arrange then the "computable" functions in a list
> 
>  F_1
>  F_2
>  .
>  .
>  .
>  F_n

	May be I am missing something obvious here, but how does
"define some notion of computability" ==> "Computable functions can be
  arranged in a list"?
	Isn't it possible to have a definition of computability that
creates a set of computable functions of aleph-1 cardinality (ie., a
set with uncountably infinite elements)?
	Is there a restricted concept of "definition (in sci.logic
terms, since comp.theory would just say the concept of "definition"
is itself ill-defined; cross-posting makes life difficult) which
limits things to recursive-enumerability?

Ajoy.
0
ajoyk (16)
1/18/2005 3:39:34 AM
Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:

> Isn't it possible to have a definition of computability that
> creates a set of computable functions of aleph-1 cardinality (ie., a
> set with uncountably infinite elements)?

  No doubt such a concept can be produced, but Turing, G�del, Church
etc were concerned with functions from N to N computable by an
algorithm.
0
torkel (478)
1/18/2005 3:55:42 AM
Torkel Franzen wrote:

> Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:
> 
> 
>>Isn't it possible to have a definition of computability that
>>creates a set of computable functions of aleph-1 cardinality (ie., a
>>set with uncountably infinite elements)?
> 
> 
>   No doubt such a concept can be produced, but Turing, G�del, Church
> etc were concerned with functions from N to N computable by an
> algorithm.

    Yes, each computable function is computable with an
algorithm (in other words, recursive), but the set of computable
functions would be uncountably infinite. How do we get from that
to ....

 >There's an apparent obstacle to formulating an absolute notion of 
 >computability. To see this assume we have defined some notion of 
 >"computability". Arrange then the "computable" functions in a list

 > F_1
 > F_2
  .
  .
  .
 > F_n

	In particular, how we can conclude that defining a notion
of computability would automatically enable us to arrange computable
functions in a list? Note that "defining" a set doesn't automatically
imply an effective procedure for checking membership in a set, at
least not in common parlance (and, as I mentioned, for comp.theory,
the word "definability" has no real formal meaning).

Ajoy.
0
ajoyk (16)
1/18/2005 4:23:44 AM
Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:

> Yes, each computable function is computable with an
> algorithm (in other words, recursive), but the set of computable
> functions would be uncountably infinite.

  There are only countably many algorithms.
0
torkel (478)
1/18/2005 4:38:56 AM
Torkel Franzen wrote:
> Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:
> 
> 
>>Yes, each computable function is computable with an
>>algorithm (in other words, recursive), but the set of computable
>>functions would be uncountably infinite.
> 
> 
>   There are only countably many algorithms.

	You have ignored the second part - there is no requirement
that a definition of a set provide an algorithm for determining
membership in the set. The set of computable functions is one such
set (ie., one with a valid definition but no algorithmic way to
validate membership). If your argument is that a "definition" is
meaningful only if it is represented by a sound algorithm, then, well,
that is a matter of perspective (it would, of course, rule out a lot
of interesting definitions, though).

Ajoy.

0
ajoyk (16)
1/18/2005 4:49:35 AM
Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:

>You have ignored the second part - there is no requirement
>that a definition of a set provide an algorithm for determining
>membership in the set.

  I take it you've set aside the idea of an uncountably infinite set
of computable functions. As for decidability, as pointed out by Aatu,
that was the great thing: the partial computable functions are
effectively enumerable, but we are saved from computably diagonalizing
out of the the computable functions by undecidability.


0
torkel (478)
1/18/2005 8:06:52 AM
Ajoy K Thamattoor wrote:

> Torkel Franzen wrote:
>>
>>   There are only countably many algorithms.
>  
>     You have ignored the second part - there is no requirement
> that a definition of a set provide an algorithm for determining
> membership in the set. The set of computable functions is one such
> set (ie., one with a valid definition but no algorithmic way to
> validate membership). If your argument is that a "definition" is
> meaningful only if it is represented by a sound algorithm, then, well,
> that is a matter of perspective (it would, of course, rule out a lot
> of interesting definitions, though).

Torkel has said nothing to indicate that this were his view. The point 
of my post was not that the set of total recursive functions is 
recursive, which it of course is not. Rather I wished to illustrate why 
it was such a surprise that there is a mathematically precise and stable 
definition of computability. This is why I spoke of an _apparent_ 
obstacle, the obstacle being that it was not clear how one could have a 
definition of computability which would not cover all intuitively 
mechanically computable functions due to the possibility of diagonalization.

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
0
1/18/2005 11:26:48 AM
Ajoy K Thamattoor says...
>
>Torkel Franzen wrote:
>> Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:
>> 
>> 
>>>Yes, each computable function is computable with an
>>>algorithm (in other words, recursive), but the set of computable
>>>functions would be uncountably infinite.
>> 
>> 
>>   There are only countably many algorithms.
>
>	You have ignored the second part - there is no requirement
>that a definition of a set provide an algorithm for determining
>membership in the set. The set of computable functions is one such
>set (ie., one with a valid definition but no algorithmic way to
>validate membership). If your argument is that a "definition" is
>meaningful only if it is represented by a sound algorithm, then, well,
>that is a matter of perspective (it would, of course, rule out a lot
>of interesting definitions, though).

I think you are misunderstanding Torkel. According to the usual meaning of
"computable function", a function is computable if and only if there is
an algorithm for computing it. There are only countably many computable
functions, since there are only countably many different algorithms.

A broader notion of defining a function is to allow a formula Phi(x,y)
define a function, provided that for each possible value of x, there is
exactly one value of y making Phi(x,y) true. Since there are only
countably many formulas, there are only countably many definable
functions, as well.

The most general notion of function from a set A to a set B is a
set F of ordered pairs <x,y> such that for every x there is exactly
one y such that <x,y> is in F. F need not be defined by a formula.
If A and B are infinite sets, then (according to ZFC) there are
uncountably many such functions, but only countably many of them
are definable.

--
Daryl McCullough

0
daryl5382 (108)
1/18/2005 11:58:15 AM
In article <351hjtF4ii8kjU1@news.dfncis.de>,
Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>Not at all strange psychologically, or to force technical meaning on
>this, modally.
>
>Back to the sqrt(2) example, what is the mathematical meaning of its
>irrationality, -if one erases from our knowledge, its proof and/or
>truth value-? We can surely use it hypothetically (it is well-formed).
>I feel stuck in the "proof says it all"/"formality" vein, and I am
>looking for some meaning outside of that point of view.

I guess this all comes down to the fact that I have no clue at all about
what you mean by "meaning."  Your use of the term differs so far from
anything I've ever seen, either in commonsense discourse or in philosophical
writing, that I can't even guess what you mean by it.  In particular, I
don't understand what "psychologically" means, or what proof or truth
has to do with meaning.

Suppose I'm learning French and someone says, "Il pleut" and I ask,
"What does that mean?"  The answer I want is that "Il pleut" means
"It's raining."  Once I know this, I know the meaning of "Il pleut."
It is entirely irrelevant whether in fact it is raining, or whether
I have any evidence that it is raining.  Even if I have no way of
telling whether or not it's in fact raining, this will not impede my
ability to learn the meaning of "Il pleut."  Nor will I need to rewrite
my phrasebook once I find out that in fact it *is* raining.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/18/2005 4:06:08 PM
In article <3523m0F4e1mo3U1@news.dfncis.de>,
Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>Whether the author is faithful to Dummett or not, and whether 
>Dummett's theory is more justifiable than Davidson's or not, it does
>seem to present a situation where -somebody- (not Dummett) supports
>the theory that some sort of mathematical truth is involved with some
>sort of mathematical meaning.

"Involved" is too vague here; that's what's getting you into trouble.

The theory here is that to know the meaning of something is to know the
conditions under which it *would* be true.  In that sense, truth is
"involved."  But to understand which conditions *would need to hold* to
make something true is entirely different from knowing *whether* it is
true, or having any grasp of how to *show* that it is true.

To know what "It is raining" means, I need to know that this sentence
*would* be true *if* little drops of water were to fall out of the sky.
I don't need to know whether in fact little drops are water *are* falling
out of the sky, let alone any *justification* of the claim that they are.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/18/2005 4:11:26 PM
tchow@lsa.umich.edu wrote:
> Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
> 
>>Not at all strange psychologically, or to force technical meaning on
>>this, modally.
>>
>>Back to the sqrt(2) example, what is the mathematical meaning of its
>>irrationality, -if one erases from our knowledge, its proof and/or
>>truth value-? We can surely use it hypothetically (it is well-formed).
>>I feel stuck in the "proof says it all"/"formality" vein, and I am
>>looking for some meaning outside of that point of view.
> 
> I guess this all comes down to the fact that I have no clue at all about
> what you mean by "meaning."  Your use of the term differs so far from
> anything I've ever seen, either in commonsense discourse or in philosophical
> writing, that I can't even guess what you mean by it.  

Fair enough. I think I can describe what I ..uh.. mean by it.
And I think I can try to say what I think you mean by it.

> In particular, I
> don't understand what "psychologically" means,

By that I take the "meaning" of an statement to be that of an
utterance by an individual in a particular situation. Your example of
someone giving directions; the meaning of such utterances depends on
context (is the speaker trustworthy? do the speaker and hearer refer
to the same things?). For the sqrt(2) example, the meaning will
involve things like possible deductions ("incommensurable values
exist (namely sqrt(2)", "you can't calculate sqrt(2) using a finite
set of integer operations"), in addition to things like "I believe
'sqrt(2) is irrational' is true" or "I heard this statement and I'm
repeating it because I trust who said it before". If I must put this
more succinctly I would say "semantic connections" or "intentions".

I can easily see that you (or Torkel) might consider most of these
things to be irrelevant for a mathematical cconcept of meaning.

> or what proof or truth has to do with meaning.

I agree with Torkel that it is uncontroversial that "truth is
involved with meaning", if their meanings are both watered down
terribly. And so I find it hard to believe that anyone would deny
this connection (like you just did, or Torkel at one point did)
unless there is a technical meaning of "meaning" that you are
demanding for our context.

> Suppose I'm learning French and someone says, "Il pleut" and I ask,
> "What does that mean?"  The answer I want is that "Il pleut" means
> "It's raining."  Once I know this, I know the meaning of "Il pleut."
> It is entirely irrelevant whether in fact it is raining, or whether
> I have any evidence that it is raining.  Even if I have no way of
> telling whether or not it's in fact raining, this will not impede my
> ability to learn the meaning of "Il pleut."  Nor will I need to rewrite
> my phrasebook once I find out that in fact it *is* raining.

So I take this analogy to mean that you (and the common sense and/or
philosophical concept) consider "meaning" to be simply translation,
or an assignment, or (I suppose more technically), a model. (I am 
certainly using the word "meaning" in such a way in many places).

I see now from your analogy how neither truth nor proof (technically)
might have nothing to do with (your intended technical meaning of)
meaning (and changing the truth won't change the meaning) that, but
it seems...unfilling to me, it

Do you really mean that that the meaning of "sqrt(2) is irrational"
is given by its translation into the definitions of all those terms?
Where does the translation stop? Don't the atomic parts of the formal
language have meaning?

But given this now, I am still unsure what it is then that you take to 
be the meaning of "sqrt(2) is irrational".
What about "sqrt(4) is irrational" (sic)?
What about Goldbach's conjecture? (as Torkel pointedly mentioned)

What is this technical meaning of "meaning" that I am missing?
Is it just translation/assignment/morphism/formalization?

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/18/2005 5:02:15 PM
tchow@lsa.umich.edu wrote:
> In article <3523m0F4e1mo3U1@news.dfncis.de>,
> Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
> 
>>Whether the author is faithful to Dummett or not, and whether 
>>Dummett's theory is more justifiable than Davidson's or not, it does
>>seem to present a situation where -somebody- (not Dummett) supports
>>the theory that some sort of mathematical truth is involved with some
>>sort of mathematical meaning.
> 
> "Involved" is too vague here; that's what's getting you into trouble.
> 
> The theory here is that to know the meaning of something is to know the
> conditions under which it *would* be true.  In that sense, truth is
> "involved."  But to understand which conditions *would need to hold* to
> make something true is entirely different from knowing *whether* it is
> true, or having any grasp of how to *show* that it is true.
> 
> To know what "It is raining" means, I need to know that this sentence
> *would* be true *if* little drops of water were to fall out of the sky.
> I don't need to know whether in fact little drops are water *are* falling
> out of the sky, let alone any *justification* of the claim that they are.

OK, now I think I know what is meant by "how it can be proved
(knowing the conditions under which it would be true)" as
different from "it can be proved (or disproved)" or "here is the
proof (or disproof)").

But I now (still?) have trouble conceiving of "knowing conditions
for it to be true" distinct from "knowing whether it is true". as in 
the sqrt(2) example.
-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/18/2005 5:18:27 PM
On 18 Jan 2005 03:58:15 -0800, Daryl McCullough <daryl@atc-nycorp.com> said:
> ...
> The most general notion of function from a set A to a set B is a set F
> of ordered pairs <x,y> such that for every x 

in A

> there is exactly one y

in B

> such that <x,y> is in F.

Pedantically &c,

Chris Menzel

0
cmenzel (185)
1/18/2005 6:14:18 PM
In article <354tonF4crug9U1@news.dfncis.de>,
Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>By that I take the "meaning" of an statement to be that of an
>utterance by an individual in a particular situation. Your example of
>someone giving directions; the meaning of such utterances depends on
>context (is the speaker trustworthy? do the speaker and hearer refer
>to the same things?).

That meaning depends on context, I agree with.  I don't think this is at the
heart of the matter.

>For the sqrt(2) example, the meaning will
>involve things like possible deductions ("incommensurable values
>exist (namely sqrt(2)", "you can't calculate sqrt(2) using a finite
>set of integer operations"), in addition to things like "I believe
>'sqrt(2) is irrational' is true" or "I heard this statement and I'm
>repeating it because I trust who said it before". If I must put this
>more succinctly I would say "semantic connections" or "intentions".
>
>I can easily see that you (or Torkel) might consider most of these
>things to be irrelevant for a mathematical cconcept of meaning.

They seem to be irrelevant for any theory of meaning, mathematical or
otherwise.  Consider the traffic example.  Again I would say that according
to common sense, we know what the directions *mean* without having any
knowledge of whether the directions are correct or not.  In fact, I have to
know what they mean before I can even begin to evaluate whether they are
correct.  Until I know what they mean, I'll be stuck on square one.  "What
did he say?  Did you catch what he said?  What did he mean by `turn
right'?"  When I finally understand what is being said, *then* I can go
about deciding whether or not it's true.

>So I take this analogy to mean that you (and the common sense and/or
>philosophical concept) consider "meaning" to be simply translation,

Not at all.  I was just using an example to illustrate a point about
meaning, namely that it does not have anything to do with whether the
statement is true or with any grounds for believing it.

>But given this now, I am still unsure what it is then that you take to 
>be the meaning of "sqrt(2) is irrational".
>What about "sqrt(4) is irrational" (sic)?

It means that if you take any two integers a and b, and then compute a^2
and 4b^2, then you will get two distinct integers.

>What about Goldbach's conjecture? (as Torkel pointedly mentioned)

It means that if you take all primes and list all pairwise sums, then every
even integer greater than 2 can be found somewhere in that list.

Surely one needs to understand the meaning of a conjecture before one can go
about looking for a proof.

Alice: "I'm trying to prove that every Hodge class on a projective
        nonsingular algebraic variety over C is a rational linear
        combination of classes of algebraic cycles."
Bob:   "What on earth does that mean?"
Alice: "Beats me.  If I knew what it meant then I'd be able to prove it."
Bob:   ??!
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/18/2005 6:36:42 PM
In article <354un3F4ffhduU1@news.dfncis.de>,
Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>But I now (still?) have trouble conceiving of "knowing conditions
>for it to be true" distinct from "knowing whether it is true". as in 
>the sqrt(2) example.

I think I addressed this in my other article, but let's take an even more
concrete example: "2^3203431780337 - 1 is prime."  I don't know if this
is true; perhaps nobody knows if it's true.  But I know what it means;
it means that if I were to divide 2^3203431780337 - 1 by any smaller
integer greater than one, I would get a nonzero remainder.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/18/2005 6:47:05 PM
Torkel Franzen wrote:
> Ajoy K Thamattoor <ajoyk@cs.stanford.edu> writes:
> 
> 
>>You have ignored the second part - there is no requirement
>>that a definition of a set provide an algorithm for determining
>>membership in the set.
> 
> 
>   I take it you've set aside the idea of an uncountably infinite set
> of computable functions.

	My mistake - in the context of this discussion (computable
function === algorithms), yes, the set will have to be countable.
I was thinking of GPACs, but there the concept of an algorithm
doesn't exist.

> As for decidability, as pointed out by Aatu,
> that was the great thing: the partial computable functions are
> effectively enumerable, but we are saved from computably diagonalizing
> out of the the computable functions by undecidability.

	Well, I would look at it the other way. We presuppose what we
want to define (basically, define computable functions as
algorithmically computable ones), and making the definition consistent
requires the introduction of undecidability.

Ajoy.


0
ajoyk (16)
1/18/2005 6:51:48 PM
But Torkel you do not explain WHY they were concerned only with integer
functions.

I give you a reason: physicalism.

The reason why "continuum machines" are not taken seriously is  because
they are impossible to build. That simple.
Who's a realist here?

Regards,

--
Eray Ozkural

0
examachine (384)
1/18/2005 8:34:41 PM
 <tchow@lsa.umich.edu> wrote:
>Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>
>>If I must put this
>>more succinctly I would say "semantic connections" or "intentions".
>>
>>I can easily see that you (or Torkel) might consider most of these
>>things to be irrelevant for a mathematical cconcept of meaning.
>
>They seem to be irrelevant for any theory of meaning, mathematical or
>otherwise.  Consider the traffic example.  Again I would say that according
>to common sense, we know what the directions *mean* without having any
>knowledge of whether the directions are correct or not.  In fact, I have to
>know what they mean before I can even begin to evaluate whether they are
>correct.  Until I know what they mean, I'll be stuck on square one.  "What
>did he say?  Did you catch what he said?  What did he mean by `turn
>right'?"  When I finally understand what is being said, *then* I can go
>about deciding whether or not it's true.

Hmm..then how do you analyse the knights/knaves statements?
When a knave (always tells falsehoods) says "to get to town, take the left 
fork", I take that statement to really mean "do not take the left fork".

(sorry if I am being obtuse, not getting the point)

>>So I take this analogy to mean that you (and the common sense and/or
>>philosophical concept) consider "meaning" to be simply translation,
>
>Not at all.  I was just using an example to illustrate a point about
>meaning, namely that it does not have anything to do with whether the
>statement is true or with any grounds for believing it.
>
>>But given this now, I am still unsure what it is then that you take to 
>>be the meaning of "sqrt(2) is irrational".
>>What about "sqrt(4) is irrational" (sic)?
>
>It means that if you take any two integers a and b, and then compute a^2
>and 4b^2, then you will get two distinct integers.
>
>>What about Goldbach's conjecture? (as Torkel pointedly mentioned)
>
>It means that if you take all primes and list all pairwise sums, then every
>even integer greater than 2 can be found somewhere in that list.

To me, both of these sound -exactly- like translation. So I don't get 
your "not at all".

Also, they both sound, as with the "il pleut/it's raining" example, very 
circular, that is one could ask what your statements mean and then give 
very legitimately the original.

>Surely one needs to understand the meaning of a conjecture before one can go
>about looking for a proof.

informally I get that but ... is that meaning beyond the translation into 
constituent formalized parts?  

>Alice: "I'm trying to prove that every Hodge class on a projective
>        nonsingular algebraic variety over C is a rational linear
>        combination of classes of algebraic cycles."
>Bob:   "What on earth does that mean?"
>Alice: "Beats me.  If I knew what it meant then I'd be able to prove it."
>Bob:   ??!

:) OK, that makes sense to me but I don't know how to reconcile it with 
your examples before.

Mitch
0
harrisq (267)
1/18/2005 9:46:13 PM
tchow@lsa.umich.edu wrote:
> Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
> ...
> There is hardly anything in the world that we know with greater certainty
> than things we have proved mathematically.

I agree. I didn't say "Suppose that someone were to find a contradiction
in ZFC," you did.

>>When someone says "sqrt(2) is irrational" it may be a bit ambiguous. Is it 
>>a statement about their intuitive (but imprecise) idea of the reals, or 
>>about the much more exactly defined objects of a formal system?
> 
> It need not be a statement about the reals; indeed, its most natural
> reading is equivalent to, "Given any strictly positive integers a and b,
> the integers a^2 and 2b^2 are distinct."  This statement is about as precise
> a statement as there is in this world.

You would think so.

"a^2 + 2b^2 has no roots with in the nonzero integers" *looks* unambiguous. 
I happen to know that if you define the integers using the normal axioms, 
it is always true.

But there is a statement of the same form - "The polynomial P has no roots 
in the integers" - which is independent of the axioms of set theory.

It *looks* just as precise.

Is it ambiguous?

If by "the integers" you mean the things that follow the axioms of the 
integers, just as when you say "a group" you mean something that follows 
the axioms of group theory, then it is.

When you say "the integers" do you mean the integers that don't include any 
roots of P, or the ones that do?

Or is there something unique called *the* integers for which all such 
statements are absolutely true or false?

Suppose mathematicians decided to accept that P has integer roots as an 
axiom. Would they know something they didn't know before? Could they be 
wrong? Or would they just be more precise in defining the integers?

>  If you regard this statement as
> imprecise, then the definition of a "formal system" is also imprecise,
> since formal systems are defined in terms of things like "symbols," "rules,"
> "strings," and so forth, which are no more precise than "integers."

But those definitions don't depend on every possible property of the 
integers, so the ambiguity does not matter.

Some statements (such as "sqrt(2) is irrational") mean the same thing 
regardless of which kind of integers you you are talking about.

Ralph Hartley
0
hartley (156)
1/18/2005 10:19:34 PM
In article <355ed5F4ierkbU1@news.dfncis.de>,
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote:
>Hmm..then how do you analyse the knights/knaves statements?
>When a knave (always tells falsehoods) says "to get to town, take the left 
>fork", I take that statement to really mean "do not take the left fork".

I regard this as something of an artificial example, whose analysis may
obscure rather than clarify "normal" examples, but if you insist on
analyzing it, I would proceed as follows.  You can analyze the situation
in two ways.  The more natural (to me) analysis is that the statement "To
get to town, take the left fork" means "To get to town, take the left
fork."  You assimilate this meaning, and then apply your knowledge of
how knaves behave in order to decide that it is false.

Alternatively, you *could* describe the situation by saying that a knave
actually *means* "do not take the left fork" when he says "take the left
fork," and that knaves always tell the truth (this is why I think
this is unnatural, but I'm trying to find an analysis that matches your
statement about what you think the knave "really means").  Then what
you're doing is to first use your knowledge that knaves assign nonstandard
meanings to sentences in order to figure out what the knave means, and
then you're using your knowledge that knaves always tell the truth to
decide that you shouldn't take the left fork.

Either way, determining the meaning is distinct from evaluating the truth.

>>>What about Goldbach's conjecture? (as Torkel pointedly mentioned)
>>
>>It means that if you take all primes and list all pairwise sums, then every
>>even integer greater than 2 can be found somewhere in that list.
>
>To me, both of these sound -exactly- like translation. So I don't get 
>your "not at all".

If a child asks me what something means, it's typically because he doesn't
understand something or has encountered new vocabulary.  To respond, I must
try to explain the unfamiliar in terms of what is already familiar.  For
example, "A `bachelor' is an unmarried man."  Now, to someone who already
knows both what a bachelor is and what an unmarried man is, this may look
like translation.  However, it conveys new information to the child.  The
reason you're having trouble with "sqrt(4) is irrational" is that you
*already* know what it means (as opposed to a child who doesn't know what
"sqrt" and "irrational" mean).  So perhaps I shouldn't even have tried to
respond the way I would respond to someone who honestly doesn't know what
"sqrt(4) is irrational" means.  It was, however, my best idea at the time
that I posted it.  By showing you how I would respond to the question if
it were asked "honestly," I hoped to illustrate to you that the question
can be answered without any reference to truth or proof.

You asked questions about infinite regress.  These are interesting
questions, but they are not objections per se (if you were thinking
of them as objections).  We can also ask how children ever learn language
since it seems we need to explain language in terms of things they already
know, and they could never get started.  How children learn language in
spite of this apparent infinite regress is interesting, but doesn't of
course prove that children never in fact learn language.

>>Surely one needs to understand the meaning of a conjecture before one can
>>go about looking for a proof.
>
>informally I get that but ... is that meaning beyond the translation into 
>constituent formalized parts?  

I don't fully understand the question.  My point is that you need to
understand a conjecture before you can even address the question of its
truth.  In practice, one way to achieve that understanding is to have
someone "translate" the conjecture into its constituent parts, because
often you will already know the meaning of the parts and so you can then
apply that knowledge to learn the meaning of the unfamiliar phrase.  But
it's clear that this translation can't be a theory of meaning by itself,
as I've been maintaining all along, and as you've pointed out with your
infinite-regress observation.

>>Alice: "I'm trying to prove that every Hodge class on a projective
>>        nonsingular algebraic variety over C is a rational linear
>>        combination of classes of algebraic cycles."
>>Bob:   "What on earth does that mean?"
>>Alice: "Beats me.  If I knew what it meant then I'd be able to prove it."
>>Bob:   ??!
>
>:) OK, that makes sense to me but I don't know how to reconcile it with 
>your examples before.

This one probably makes sense to you because you don't know the meaning of
those mathematical terms, and so you can put yourself in Bob's position
and "honestly" ask what it means.  Whereas in the case of "sqrt(4) is
irrational" you *do* know what the sentence means (after all, you came
up with it yourself to illustrate something) and so you have trouble
asking the "what does it mean" question "honestly."  If you can put
yourself into the position of a child, then you should see that the
examples are parallel.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/18/2005 11:01:59 PM
harrisq@tcs.inf.tu-dresden.de (Mitch Harris) wrote in
news:355ed5F4ierkbU1@news.dfncis.de: 

> Hmm..then how do you analyse the knights/knaves statements?
> When a knave (always tells falsehoods) says "to get to town, take the
> left fork", I take that statement to really mean "do not take the left
> fork". 
> 
> 

But it doesn't really mean that and you don't really take it that way.  If 
the knave's statement really meant to you, the opposite of what it seems to 
mean then you would take the left fork anyway, because he's lying and you 
think his words mean "take the right fork".  Once you know that the knave 
is lying, you take the right fork *because* you understand the meaning of 
his (falsely uttered) words, not because you understand his words to mean 
the opposite of what they really do mean to you.
0
nobody5290 (97)
1/18/2005 11:32:08 PM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> I happen to know that if you define the integers using the normal axioms, 
> it is always true.

  What do you mean by "define the integers using the normal axioms"?
What definition is that?

> If by "the integers" you mean the things that follow the axioms of the 
> integers, just as when you say "a group" you mean something that follows 
> the axioms of group theory, then it is.

  This is an extremely mysterious statement. How would you apply your
on the face of it baffling line of thought to the case of "ZFC is
consistent". Is it "ambiguous" in your sense?
0
torkel (478)
1/19/2005 6:40:18 AM
Kenneth Doyle wrote:
> harrisq@tcs.inf.tu-dresden.de (Mitch Harris) wrote 
> 
>>Hmm..then how do you analyse the knights/knaves statements?
>>When a knave (always tells falsehoods) says "to get to town, take the
>>left fork", I take that statement to really mean "do not take the left
>>fork". 
> 
> But it doesn't really mean that and you don't really take it that way.  If 
> the knave's statement really meant to you, the opposite of what it seems to 
> mean then you would take the left fork anyway, because he's lying and you 
> think his words mean "take the right fork".  Once you know that the knave 
> is lying, you take the right fork *because* you understand the meaning of 
> his (falsely uttered) words, not because you understand his words to mean 
> the opposite of what they really do mean to you.

That makes sense. It seems best to separate the meaning of the formal
statement (the surface meaning?) from inferences you can make with it 
given a particular context.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/19/2005 10:01:42 AM
tchow@lsa.umich.edu wrote:
> In article <1105780027.473885.47200@c13g2000cwb.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> >Paraphrasing,  amongst other things, Tim Chow asked if AC and the
other
> >axioms of ZF are true.  Unless he was using a non-model-theoretic
use
> >of the term "true," the ZFC is just as true as the group axioms,
since
> >we can exhibit models for both sets of axioms.
>
> I used the term the way mathematicians normally use them, which never
causes
> problems except when someone suddenly decides to get skeptical and
ask,
> following Pontius Pilate, "What is truth?"  For the present purposes,
I
> can eliminate all uses of the word "true" by simply replacing the
statement
> of which I am predicating truth with the statements themselves.  This
is
> cumbersome, but is helpful psychologically for those who aren't
practiced
> in doing such things themselves.  So when I ask if AC is true, I am
asking
> if the Cartesian product of nonempty sets is nonempty.

Where, ZF or ZFC?  Remember, models of ZF + "not AC" exist.  Without
this crucial bit of information, the question has no (single -- I see
four) answer.

>When I ask if "ZFC
> is consistent" is true, I am asking if ZFC is consistent.  When I ask
if
> "`Every vector space has a basis' is provable in ZFC" is true, I am
asking
> if `Every vector space has a basis' is provable in ZFC.
>
> So if you claim not to understand what I mean when I say that "ZFC is
> consistent" is true, even after you understand how to eliminate the
word
> "true" as I have just demonstrated, then you are really claiming not
> to understand what I mean when I say that ZFC is consistent.  And of
> course, in this example, you *do* understand what I mean, because you
> make similar assertions yourself, like "ZFC is consistent if and only
if
> it has models," which you presumably wouldn't make if you didn't know
> what such an assertion meant, and which presupposes that you know
what
> it means for ZFC to be consistent.

I didn't say that ZFC is consistent iff it has models.  (Though that's
obviously true).  What I did say is that ZFC is true relative to a
fixed structure S iff S is a model for ZFC.  It is this relativity I
wish to emphasize, since it captures some of the context sensitivity
inherent in mathematical work.  For instance, If S is the dihedral
group, S is not a model for ZFC.  A group theorist, while working with
D_8, has no interest in the truth of ZFC.  Why postulate truths that
don't matter?  One would have to make many objectionable ontological
commitments to support these sorts of Platonic truths.

> The difference between ZFC and the axioms for group theory is not any
> kind of interesting structural difference between the first-order
theories
> themselves, as you point out.  The difference is that ZFC is usually
used
> in the context of trying to capture certain features of general
mathematical
> discourse---in particular, statements that we make all the time that
we feel
> we understand the meaning of unambiguously.

If you "all" feel that you can understand the meaning of the
relationships between different sets *intuitively* and unambiguously,
the truth of AC in the sense you describe above is trivial -- just tell
me if your intuitive notion allows a choice function.  Brouwer's
intuitive notion didn't.  You must be careful when speaking for all
mathematicians.

>Although maybe you have not
> articulated it to yourself explicitly, I bet you in fact believe that
you're
> making a specific, meaningful statement when you say "`Every vector
space
> has a basis' is provable in ZFC."  <snip>

Certainly -- and in contexts in which it's clear I'm working with ZFC,
the statement "Every vector space has a basis" is just as meaningful.
But stripped of it's context, this second statement loses obvious
meaning.  I'd have to explain what I meant, coherently, without
assuming that the meaning is obvious (since it isn't).

> The group-theoretic axioms, however, are not introduced in order to
capture
> statements that mathematicians assert directly without any sense of
> ambiguity.  Mathematicians do not go around saying things like, "For
every
> x, y, and z, (x * y) * z = x * (y * z)" without further explanation
about
> what x, y, and z are and what * is.
>

Why is the notion of sets and relations more natural than the notion of
a Galois correspondence?  Both occur in the "same place"  -- ie, have
the same ontological status -- after all.  Historically, the notion of
a group was introduced to capture the behavior of certain sets of
automorphisms of roots of polynomials.  In fact, one could easily argue
that the group axioms are *more* in line with the intended model than
ZFC, since they obviously satisfy their historical motivations, whereas
it is demonstrable that different notions of sets and elementness
exist.

'cid 'ooh

0
poopdeville (133)
1/19/2005 11:49:36 AM
Ralph Hartley says...

>"a^2 + 2b^2 has no roots with in the nonzero integers" *looks* unambiguous. 
>I happen to know that if you define the integers using the normal axioms, 
>it is always true.
>
>But there is a statement of the same form - "The polynomial P has no roots 
>in the integers" - which is independent of the axioms of set theory.
>
>It *looks* just as precise.
>
>Is it ambiguous?
>
>If by "the integers" you mean the things that follow the axioms of the 
>integers, just as when you say "a group" you mean something that follows 
>the axioms of group theory, then it is.
>
>When you say "the integers" do you mean the integers that don't include any 
>roots of P, or the ones that do?
>
>Or is there something unique called *the* integers for which all such 
>statements are absolutely true or false?

If a statement Phi in the language of arithmetic is independent
of the axioms of Peano Arithmetic, then that means that there exists
a model of PA + Phi and there also exists a model of PA + ~Phi. However,
the two models are *not* both models of the naturals (I assume you
meant the naturals, rather than the integers, not that it makes much
difference). It means that at least one of the models is nonstandard,
meaning that it contains elements that are not naturals.

It isn't at all *ambiguous* that a model is nonstandard; all nonstandard
models of PA have features that are lacking in the standard model. In
particular, every nonstandard model contains a standard submodel. So if
Phi is independent of PA, then either

     1. Every model of PA+Phi has a submodel that is a model of PA+~Phi.
(in which case, Phi is false in the standard model)
or
     2. Every model of PA+~Phi has a submodel that is a model of PA+Phi.
(in which case, Phi is true in the standard model).

So, it isn't that Phi is *ambiguous*. We just don't
know whether it is true, or not. Those aren't the same thing.
If I give you a locked safe containing an amount of money,
it isn't *ambiguous* whether the safe contains more than
$1 million. You just don't know whether it's true.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/19/2005 12:00:04 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote in
news:356pg6F4jmllhU1@news.dfncis.de: 

> Kenneth Doyle wrote:
>> harrisq@tcs.inf.tu-dresden.de (Mitch Harris) wrote 
>> 
>>>Hmm..then how do you analyse the knights/knaves statements?
>> 
>> Once you know that the knave is lying, you take the right
>> fork *because* you understand the meaning of his (falsely uttered)
>> words, not because you understand his words to mean the opposite of
>> what they really do mean to you. 
> 
> That makes sense. It seems best to separate the meaning of the formal
> statement (the surface meaning?) from inferences you can make with it 
> given a particular context.
> 

I'm not sure what you're getting at.  What I see is that truth "propagates" 
through a chain of inference by virtue of the truth-functional design of a 
logical system.
 
In the version of the knights/knaves problem that I encountered, the puzzle 
is solved by asking one of them what the other would answer were we to ask 
the question of the other.  By getting one of them to answer for the other, 
we eliminate the need to know which one is telling the truth because they 
both know that about each other and that knowledge is conveyed in their 
answer, regardless of whether or not they are telling the truth when they 
tell us how the other would respond; if you see what I mean.  Note that we 
can confidently chose the correct fork, without ever knowing which is the 
knight and which the knave. 

0
nobody5290 (97)
1/19/2005 12:05:08 PM
tchow@lsa.umich.edu wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote:
> 
>>Hmm..then how do you analyse the knights/knaves statements?
>>When a knave (always tells falsehoods) says "to get to town, take the left 
>>fork", I take that statement to really mean "do not take the left fork".
> 
> I regard this as something of an artificial example, whose analysis may
> obscure rather than clarify "normal" examples, but if you insist on
> analyzing it, I would proceed as follows.  

As artificial as, say, "the sqrt(4) is irrational", or Goldbach's
conjecture. But your ways of answering do clarify the situation
(somewhat) for me.

> You can analyze the situation
> in two ways.  The more natural (to me) analysis is that the statement "To
> get to town, take the left fork" means "To get to town, take the left
> fork."  You assimilate this meaning, and then apply your knowledge of
> how knaves behave in order to decide that it is false.
> 
> Alternatively, you *could* describe the situation by saying that a knave
> actually *means* "do not take the left fork" when he says "take the left
> fork," and that knaves always tell the truth (this is why I think
> this is unnatural, but I'm trying to find an analysis that matches your
> statement about what you think the knave "really means").  Then what
> you're doing is to first use your knowledge that knaves assign nonstandard
> meanings to sentences in order to figure out what the knave means, and
> then you're using your knowledge that knaves always tell the truth to
> decide that you shouldn't take the left fork.
> 
> Either way, determining the meaning is distinct from evaluating the truth.

OK, this now makes it clear that meaning and truth should be treated 
distinctly (however I have a hard time articulating that distinction 
clearly, "meaning has something to do with the original statement, 
truth with inferences about the statement", ). So you have a very 
particular prescriptive (or stipulative) definition of "meaning" in 
mind when talking about mathematical statements.

....

>>>Surely one needs to understand the meaning of a conjecture before one can
>>>go about looking for a proof.
>>
>>informally I get that but ... is that meaning beyond the translation into 
>>constituent formalized parts?  
> 
> I don't fully understand the question.  My point is that you need to
> understand a conjecture before you can even address the question of its
> truth.  In practice, one way to achieve that understanding is to have
> someone "translate" the conjecture into its constituent parts, because
> often you will already know the meaning of the parts and so you can then
> apply that knowledge to learn the meaning of the unfamiliar phrase.  But
> it's clear that this translation can't be a theory of meaning by itself,
> as I've been maintaining all along, and as you've pointed out with your
> infinite-regress observation.

OK. So (you seem to be saying and I -am- saying that) translation is
an important part of a theory of meaning. But not all of it. So what
is there in addition? To "understanding" something (get its meaning),
what needs to be done after the translation, since that is not enough?

>>>Alice: "I'm trying to prove that every Hodge class on a projective
>>>       nonsingular algebraic variety over C is a rational linear
>>>       combination of classes of algebraic cycles."
>>>Bob:   "What on earth does that mean?"
>>>Alice: "Beats me.  If I knew what it meant then I'd be able to prove it."
>>>Bob:   ??!
>>
>>:) OK, that makes sense to me but I don't know how to reconcile it with 
>>your examples before.
> 
> This one probably makes sense to you because you don't know the meaning of
> those mathematical terms, and so you can put yourself in Bob's position
> and "honestly" ask what it means.  

Also, your example "means" something to people who can't translate
all the words. But I realize that kind of meaning is not part of our
current discussion.

> Whereas in the case of "sqrt(4) is
> irrational" you *do* know what the sentence means (after all, you came
> up with it yourself to illustrate something) and so you have trouble
> asking the "what does it mean" question "honestly."  If you can put
> yourself into the position of a child, then you should see that the
> examples are parallel.

OK. I got it.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/19/2005 1:27:40 PM
Kenneth Doyle wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote
>>Kenneth Doyle wrote:
>>>harrisq@tcs.inf.tu-dresden.de (Mitch Harris) wrote 
>>>
>>>>Hmm..then how do you analyse the knights/knaves statements?
>>>
>>>Once you know that the knave is lying, you take the right
>>>fork *because* you understand the meaning of his (falsely uttered)
>>>words, not because you understand his words to mean the opposite of
>>>what they really do mean to you. 
>>
>>That makes sense. It seems best to separate the meaning of the formal
>>statement (the surface meaning?) from inferences you can make with it 
>>given a particular context.
> 
> I'm not sure what you're getting at.  What I see is that truth "propagates" 
> through a chain of inference by virtue of the truth-functional design of a 
> logical system.

and (I think what Torkel and Tim are trying to convince me of) this 
should be managed as a distinct concept from the meaning of the answers.

> In the version of the knights/knaves problem that I encountered, the puzzle 
> is solved by asking one of them what the other would answer were we to ask 
> the question of the other.  By getting one of them to answer for the other, 
> we eliminate the need to know which one is telling the truth because they 
> both know that about each other and that knowledge is conveyed in their 
> answer, regardless of whether or not they are telling the truth when they 
> tell us how the other would respond; if you see what I mean.  Note that we 
> can confidently chose the correct fork, without ever knowing which is the 
> knight and which the knave. 

Right. So their answers are either "Yes" or "No". These answers have
a meaning ("Yes" = "I agree with your statement (or the positive
version of your question)"), but they also have a truth value
(correct or not), based on the relation of what utterance they gave
("yes" or "no"), their trustworthiness (the relationship between
their internal beliefs and their utterances), and which fork the town
is actually on.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/19/2005 1:49:58 PM
Ralph Hartley wrote:
> by "the integers" you mean the things that follow the axioms of the 
> integers, just as when you say "a group" you mean something that follows 
> the axioms of group theory
....
> Or is there something unique called *the* integers for which all such 
> statements are absolutely true or false?

It looks like I said (or implied) something here that is stronger than what 
I really think.

My actual position is somewhere in between.

Any model of the group axioms is a group. I don't think I, or most 
mathematicians, would accept *any* model of PA as "the integers".

The intuitive notion of "the integers" is more specific than that. There 
are models of PA that just about everyone will agree are not the "intended" 
model.

But that does not mean that it is totally precise. There may some room for 
debate about which model is intended.

For instance, when someone talks about the intended model of ZFC they don't 
mean just any model, they mean sets (the class of sets and their relations 
etc.).

But some mean the sets that *do* include an uncountable cardinal less than 
the continuum, and some mean sets that do not.

The intuitive notion of "sets" is not precise enough. Which model of ZFC do 
we mean by it? Is it a model for which CH is true, or one in which it is false?

I don't conclude that the concept of "sets" is ambiguous just because CH is 
independent of ZFC, but because of that *and* the fact that (reasonable) 
mathematicians disagree about whether CH is true.

In this view deciding to accept or reject CH amounts to refining the 
intuitive idea of sets.

It should come as no surprise that such refinement is needed. Ancient 
hunter gatherers may have had a notion of sets, but to ask if AC (to say 
nothing of CH) was true of that notion is just silly. They (or 90%+ of 
living humans for that matter) would have no idea what you were talking about.

When I start answering my own posts, that usually means it's time to quit.

Ralph Hartley
0
hartley (156)
1/19/2005 2:02:51 PM
Ralph Hartley <hartley@aic.nrl.navy.mil> writes:

> But that does not mean that it is totally precise. There may some room for 
> debate about which model is intended.

  There is always theoretical "room" for debate about anything, but
there is no in fact any debate in any mathematical context about what
the natural numbers are.

> But some mean the sets that *do* include an uncountable cardinal less than 
> the continuum, and some mean sets that do not.

  Really? Where do we find this difference manifested?
0
torkel (478)
1/19/2005 2:22:33 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote in
news:3576s6F4he3b5U1@news.dfncis.de: 

> Kenneth Doyle wrote:
>> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote
>>>Kenneth Doyle wrote:
>>>>harrisq@tcs.inf.tu-dresden.de (Mitch Harris) wrote 
>>>>
>>>>>Hmm..then how do you analyse the knights/knaves statements?
>>>>
>>>>Once you know that the knave is lying, you take the right
>>>>fork *because* you understand the meaning of his (falsely uttered)
>>>>words, not because you understand his words to mean the opposite of
>>>>what they really do mean to you. 
>>>
>>>That makes sense. It seems best to separate the meaning of the formal
>>>statement (the surface meaning?) from inferences you can make with it
>>> given a particular context. 
>> 
>> I'm not sure what you're getting at.  What I see is that truth
>> "propagates" through a chain of inference by virtue of the
>> truth-functional design of a logical system. 
> 
> and (I think what Torkel and Tim are trying to convince me of) this 
> should be managed as a distinct concept from the meaning of the
> answers. 
> 
>> In the version of the knights/knaves problem that I encountered, the
>> puzzle is solved by asking one of them what the other would answer
>> were we to ask the question of the other.  By getting one of them to
>> answer for the other, we eliminate the need to know which one is
>> telling the truth because they both know that about each other and
>> that knowledge is conveyed in their answer, regardless of whether or
>> not they are telling the truth when they tell us how the other would
>> respond; if you see what I mean.  Note that we can confidently chose
>> the correct fork, without ever knowing which is the knight and which
>> the knave. 
> 
> Right. So their answers are either "Yes" or "No". These answers have
> a meaning ("Yes" = "I agree with your statement (or the positive
> version of your question)"), but they also have a truth value
> (correct or not), based on the relation of what utterance they gave
> ("yes" or "no"), their trustworthiness (the relationship between
> their internal beliefs and their utterances), and which fork the town
> is actually on.
> 

Their trustworthiness has nothing to do with it, provided we ask the right 
question.  If we ask either of them, "which road will the other point to if 
I ask him the way to town?" then both will point to the same road.  We want 
to take the other road regardless of whether the knight is answering for 
the knave, or the knave is answering for the knight.
0
nobody5290 (97)
1/19/2005 2:30:10 PM
In article <3575icF4hrpshU1@news.dfncis.de>,
Mitch Harris  <harrisq@tcs.inf.tu-dresden.de> wrote:
>OK. So (you seem to be saying and I -am- saying that) translation is
>an important part of a theory of meaning. But not all of it. So what
>is there in addition? To "understanding" something (get its meaning),
>what needs to be done after the translation, since that is not enough?

Now it sounds like you agree with (or at least understand) my view that
the proof of a statement is separate from its meaning, and you want to
go further and formulate a systematic theory of meaning.  I have no such
systematic theory, and the ones I've seen all have their problems.  However,
it's clear what sorts of things you would consider when you start trying to
formulate such a theory.  The meaning of simple nouns like "sun" is, or
must be closely related to, the physical object to which the word refers
(in this case, the sun).  The meaning of simple verbs must be closely
related to the action to which the verb refers.  Etc.  I'll leave you to
pursue these paths if you want, but I'm not really interested in getting
deeply into such a discussion right now.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/19/2005 2:55:37 PM
Kenneth Doyle wrote:
> Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote 
>>Kenneth Doyle wrote:
>>>Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote
>>>>Kenneth Doyle wrote:
>>>>>harrisq@tcs.inf.tu-dresden.de (Mitch Harris) wrote 
>>>>>
>>>>>>Hmm..then how do you analyse the knights/knaves statements?
>>>>>
>>>>>Once you know that the knave is lying, you take the right
>>>>>fork *because* you understand the meaning of his (falsely uttered)
>>>>>words, not because you understand his words to mean the opposite of
>>>>>what they really do mean to you. 
>>>>
>>>>That makes sense. It seems best to separate the meaning of the formal
>>>>statement (the surface meaning?) from inferences you can make with it
>>>>given a particular context. 
>>>
>>>I'm not sure what you're getting at.  What I see is that truth
>>>"propagates" through a chain of inference by virtue of the
>>>truth-functional design of a logical system. 
>>>
>>>In the version of the knights/knaves problem that I encountered, the
>>>puzzle is solved by asking one of them what the other would answer
>>>were we to ask the question of the other.  By getting one of them to
>>>answer for the other, we eliminate the need to know which one is
>>>telling the truth because they both know that about each other and
>>>that knowledge is conveyed in their answer, regardless of whether or
>>>not they are telling the truth when they tell us how the other would
>>>respond; if you see what I mean.  Note that we can confidently chose
>>>the correct fork, without ever knowing which is the knight and which
>>>the knave. 
>>
>>Right. So their answers are either "Yes" or "No". These answers have
>>a meaning ("Yes" = "I agree with your statement (or the positive
>>version of your question)"), but they also have a truth value
>>(correct or not), based on the relation of what utterance they gave
>>("yes" or "no"), their trustworthiness (the relationship between
>>their internal beliefs and their utterances), and which fork the town
>>is actually on.
> 
> Their trustworthiness has nothing to do with it, provided we ask the right 
> question.  If we ask either of them, "which road will the other point to if 
> I ask him the way to town?" then both will point to the same road.  We want 
> to take the other road regardless of whether the knight is answering for 
> the knave, or the knave is answering for the knight.

Yes, that's the point of the puzzle, to come up with a question that
doesn't depend on their trustworthiness. But the original intent of
my query "how do you analyse the knights/knaves statements?" was not
the naive "how do you show that the answer to the puzzle works
without relying on their trustworthiness?", but rather "what is the
meaning of their answers?", and as Tim pointed out, this is distinct
from the correctness (truth) of their answers.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
1/19/2005 3:35:04 PM
In article <cslp19$4b2$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>The intuitive notion of "the integers" is more specific than that. There 
>are models of PA that just about everyone will agree are not the "intended" 
>model.
>
>But that does not mean that it is totally precise. There may some room for 
>debate about which model is intended.

In the case of the integers, there really isn't any debate.  You talk about
PA, by which I assume you mean first-order Peano arithmetic; that system
certainly has many different models.  However, the *second-order* Peano
axioms admit only one model, and *that's* what people think of as the
natural numbers.

>For instance, when someone talks about the intended model of ZFC they don't 
>mean just any model, they mean sets (the class of sets and their relations 
>etc.).
>
>But some mean the sets that *do* include an uncountable cardinal less than 
>the continuum, and some mean sets that do not.

In the case of sets, there is certainly more controversy.  Some people might
describe the situation as you do.  However, the usual way of regarding the
situation is that there is agreement that the intended model of ZFC is V,
the class of all sets (not to be confused with your use of the letter "V"
to denote a weak axiomatic system), and what people aren't sure of is
whether CH (the continuum hypothesis) is true in V or not.  The disagreement
is not over meaning, but over truth.

You probably have in mind models of ZFC in which CH is true (e.g., L,
the class of all constructible sets) and models of ZFC in which CH is false
(e.g., countable transitive models of ZFC obtained by forcing; let's pick
one and call it M).  But even the people who somehow believe that V, L,
and M are all on equal footing in some sense do not go about saying that
"M might be the intended model."  It's clear that M is artificially
constructed to falsify CH, and nobody believes that this is what we meant
all along by V.  As for L, this is slightly hazier; certainly we didn't
have L specifically in mind before Goedel described it precisely, but
a minority of people feel that now that the description of L has been
articulated explicitly, it matches their imprecise intuitions about V
well enough that they are prepared to accept "V = L" as a basic axiom.

However, in any case, the mere existence of independence results doesn't
automatically mean that there is *ambiguity* about what V is.  It just
means that there are facts about V whose truth value we don't know, and
that are inaccessible to our current methods of ascertaining truths about
it.  Indeed, you yourself agree with this:

>I don't conclude that the concept of "sets" is ambiguous just because CH is 
>independent of ZFC

On the other hand you go on to say that:

>but because of that *and* the fact that (reasonable) 
>mathematicians disagree about whether CH is true.

But this isn't enough to establish ambiguity either.  Let's go back to
the case of the natural numbers.  Reasonable mathematicians disagree on
various statements about the natural numbers, too.  For example, "Measurable
cardinals are consistent with ZFC" can be formulated as purely arithmetical
statement, but it's unprovable in ZFC (and in fact in much stronger systems).
Accepting such a statement doesn't change our idea of what the natural
numbers *are*; it just adds to our knowledge of them.

>In this view deciding to accept or reject CH amounts to refining the 
>intuitive idea of sets.

That's a possible view, but the reasons you've adduced in support of it
are inadequate.

I think you'd be better off arguing directly that the cumulative hierarchy
of sets is simply not a very clear concept.  It's clear at the low
levels, but as we move further and further up, we rely more and more on
"extrapolations" of our intuitions, and so we gradually lose clarity.
While we can't necessarily point to a specific spot where things suddenly
start becoming unclear, and maybe some people's intuition carries them
further than others', eventually everyone loses clarity.  This is a much
vaguer argument, but at least it has more plausibility than the types of
arguments from independence results you've been proposing.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/19/2005 4:00:16 PM
Mitch Harris <harrisq@tcs.inf.tu-dresden.de> wrote in
news:357d18F4hsr8rU1@news.dfncis.de: 

>  We want 
>> to take the other road regardless of whether the knight is answering
>> for the knave, or the knave is answering for the knight. 
> 
> Yes, that's the point of the puzzle, to come up with a question that
> doesn't depend on their trustworthiness. But the original intent of
> my query "how do you analyse the knights/knaves statements?" was not
> the naive "how do you show that the answer to the puzzle works
> without relying on their trustworthiness?", but rather "what is the
> meaning of their answers?", and as Tim pointed out, this is distinct
> from the correctness (truth) of their answers.
> 

Right, sorry to harp on about it.

The issue does get confused when I consider a phrase like, "When I tell the 
truth, I mean what I say and when I lie, I don't".  In this sense, the 
knight means what he says when he indicates the wrong road (being truthful 
about the knave's lying).  His answer means that the knave would have 
indicated the wrong road.  The knave, on the other hand, does not mean what 
he says when he indicates the wrong road (being a liar about the kight's 
truthfulness).  His answer means that that the knight would have indicated 
the correct road.  We seem to have an identical statement meaning opposite 
things on two occasions (where nothing but the speaker has changed).  I 
would suggest that that merely is an indicator that there's some sort of 
equivocation on the word "meaning".  The sense in which I say, "His answer 
means..." is the sense where "meaning" is synonymous with "implication".  
The sense in which I say, "the knight means what he says...", simply uses 
"meaning" as a synonym for "honesty".
0
nobody5290 (97)
1/19/2005 4:31:39 PM
t...@lsa.umich.edu wrote:
> In article <1106135376.526016.178150@f14g2000cwb.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> >I didn't say that ZFC is consistent iff it has models.  (Though
that's
> >obviously true).  What I did say is that ZFC is true relative to a
> >fixed structure S iff S is a model for ZFC.
>
> I wasn't intending to argue that in fact you had made that statement,
just
> that you were happy making statements of that type.  This was an
attempt
> to show you that you are guilty of exactly the same sin you're
accusing
> me of.  In your parenthetical remark here you say, "that's obviously
> *true*" (emphasis mine).  What do you mean by true?  We have the
sentence
>
> (*)  "ZFC is consistent iff ZFC has models."
>
> You're comfortable asserting that (*) is *true*.  I ask, true in what
> model?  Model of what system?  of PA?  of ZFC?

If you wanted to be really clever, you could have asked why I think
"ZFC is consistent iff ZFC has models" when the proof of that fact
requires AC instead of this.  ;-)  Now that I know that, I'm not so
sure that "ZFC is consistent iff ZFC has models" is true.

However, let's assume that AC holds.  (So we're working within ZFC)
The class of models of ZFC, being non-empty, is a model for FOL + "ZFC
has models."  The completeness theorem gives you "ZFC is consistent" as
a consequence of FOL + "ZFC has models".  If you're asking me to
exactly pin this down, it's going to be an ugly beast.  But you can
always create a model for a theorem by throwing the theorem's
hypotheses in as axioms with FOL.


>You complained about
> "the cartesian product of nonempty sets is nonempty" as having no
fixed
> meaning, so I'm going to turn around and claim that (*) has no fixed
> meaning as far as I can see.  It means one thing if (*) is
interpreted
> in some model of ZFC where the integers are nonstandard and it means
> something else if it is interpreted in a model of ZFC in which the
> integers are the standard integers.  (If there are nonstandard
integers
> then there are nonstandard proofs, and "consistency" means that there
> aren't any proofs of a contradiction, not even nonstandard ones.)
> 

Fair enough.  I was unclear.

'cid 'ooh

0
poopdeville (133)
1/19/2005 10:10:22 PM
poopdeville@gmail.com writes:

> If you wanted to be really clever, you could have asked why I think
> "ZFC is consistent iff ZFC has models" when the proof of that fact
> requires AC instead of this.

  The proof does not require AC.
0
torkel (478)
1/19/2005 10:20:13 PM
["Followup-To:" header set to sci.logic.]
On 19 Jan 2005 23:44:47 GMT, tchow@lsa.umich.edu <tchow@lsa.umich.edu> said:
> In article <1106172622.476848.294340@z14g2000cwz.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
>>If you wanted to be really clever, you could have asked why I think
>>"ZFC is consistent iff ZFC has models" when the proof of that fact
>>requires AC instead of this.
>
> It doesn't require AC.  You might be thinking that to prove the
> completeness theorem for alphabets of arbitrary cardinality, AC
> is needed.  

Just for the thesis that all cardinals are alephs, though, right?  I
can't think off the top of my head of any place you need it in the
actual construction as long as your language is well-orderable.

He might also have been thinking of the strong form of the downward
Lowenheim-Skolem-Tarski theorem, which lives in the same general
neighborhood.

Chris Menzel

0
cmenzel (185)
1/20/2005 5:11:59 PM
Torkel Franzen wrote:
> poopdeville@gmail.com writes:
> 
> 
>>If you wanted to be really clever, you could have asked why I think
>>"ZFC is consistent iff ZFC has models" when the proof of that fact
>>requires AC instead of this.
> 
> 
>   The proof does not require AC.

I've read a little bit about W. V. Quine, and was wondering what he
said, wrote or thought on truth, knowability, reality or
unreality in mathematics.

David Bernier
0
david250 (151)
1/21/2005 5:55:02 AM
tchow@lsa.umich.edu wrote:
> In article <1106172622.476848.294340@z14g2000cwz.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> >If you wanted to be really clever, you could have asked why I think
> >"ZFC is consistent iff ZFC has models" when the proof of that fact
> >requires AC instead of this.
>
> It doesn't require AC.  You might be thinking that to prove the
> completeness theorem for alphabets of arbitrary cardinality, AC
> is needed.  But ZFC uses only a countable alphabet.

Of course.  I keep mistaking "ZFC is consistent iff ZFC has models"
with the more general "Consistency iff Satisfiability" since they're
more-or-less the same issue -- at least in context.

>
> >However, let's assume that AC holds.  (So we're working within ZFC)
> >The class of models of ZFC, being non-empty, is a model for FOL +
"ZFC
> >has models."  The completeness theorem gives you "ZFC is consistent"
as
> >a consequence of FOL + "ZFC has models".
>
> No, you're missing the point.  I'm not asking you to show me a
*proof*
> of the statement in question.  I'm asking you what it *means*.  What
does
>
>   (*) ZFC is consistent iff ZFC has models
>
> mean?  You balked at
>
>   (**) The cartesian product of nonempty sets is nonempty
>
> claiming not to know what (**) means without knowing which model of
ZFC
> to interpret it in.  Well, I'm turning around and telling you that in
that
> case, I don't know what (*) means until you tell me which model of
ZFC
> to interpret it in.  If you interpret it a model of ZFC with
nonstandard
> integers, then it means something quite different from what it means
in
> a model of ZFC with standard integers.
>
> To put it another way, now that you know AC isn't involved,
presumably
> you are again comfortable with saying that (*) is *true*.  But true
in
> what model?  Or do you mean true simpliciter?  But what is truth
> simpliciter?

It's true in every model for the satisfiability relation.  This follows
directly from the fact that a proof of "Consistency iff Satisfiability"
exists and the soundness of FOL.  Coming up with a concrete example of
such a model wouldn't be too difficult, just tedious.  

'cid 'ooh

0
poopdeville (133)
1/24/2005 1:44:28 AM
poopdeville@gmail.com writes:

> It's true in every model for the satisfiability relation.

  And what does it mean that it's true in every model for the
satisfiability relation?
0
torkel (478)
1/24/2005 6:58:09 AM
In article <vcbsm4rgvpa.fsf@beta19.sm.ltu.se>,
Torkel Franzen  <torkel@sm.luth.se> wrote:
>poopdeville@gmail.com writes:
>> It's true in every model for the satisfiability relation.
>  And what does it mean that it's true in every model for the
>satisfiability relation?

Exactly.  However, I suspect that 'cid 'ooh still doesn't see what I'm
driving at.  I will try again, although I'm beginning to lose hope.

Recall 'cid 'ooh's objection to my usage of "true": What does it mean
to say that "the cartesian product of nonempty sets is nonempty" is
true?  My response was, following Tarski, that I was using "true" in an
eliminable way, and that by saying "the cartesian product of nonempty
sets is nonempty" is true, I was saying no more than that the cartesian
product of nonempty sets is nonempty.  'cid 'ooh balked at this,
claiming not to understand what it means for the cartesian product of
nonempty sets to be nonempty unless I specified a model that it was
true *in*.

My response was that 'cid 'ooh was guilty of exactly the same sin he was
charging me with, namely asserting mathematical statements flat out without
saying which model of ZFC they were true *in*.  As an example, I gave the
statement

   (*) ZFC is consistent iff ZFC has models.

I asked, what does this mean?  I don't understand what it means until you
tell me which model of ZFC it's true *in*.  (Of course, I do in fact know
what it means, but am asking these questions to illustrate my point that
'cid 'ooh's objections can be turned against him.)  'cid 'ooh's reaction
was, essentially, that (*) is true in all models of ZFC.

But this is not an answer at all.  If it were an answer, then I could
respond to the question of why I accept that "the cartesian product of
nonempty sets is nonempty" by simply observing that it's true in all
models of ZFC.  That's correct, but is obviously not going to satisfy
'cid 'ooh.  Any axiom is true in all models of that axiom; this trivial
fact nothing about whether to accept it, and of course is not why I
accept it.

'cid 'ooh has a blind spot; when others make mathematical assertions,
such as "every vector space has a basis," he claims not to be able to
understand any mathematical statement as having a specific, definite
meaning unless some model is explicitly specified that it is true *in*.
Yet he himself makes mathematical assertions like (*) and like

  (**) Theorems of ZFC, e.g., (*), are true in all models of ZFC

baldly, meaning something specific and definite by them, without saying
which model of these statements are true *in*.  When pressed, he says
that they are true in all models, which obviously is insufficient to
fix a specific, determinate meaning for the statements in question, and
provides no basis for accepting or rejecting (*) or (**).

What 'cid 'ooh needs to acknowledge is that he makes lots of mathematical
statements flat out without needing to say which model they're "true in."
As Torkel Franzen points out, an example is (**); surely it cannot be the
case that (**) makes no sense unless one says which model it's true in,
or else we get an infinite regress (which model is "(**) is true in model X"
true in?).

Once that is acknowledged, then maybe we can get back to the other
questions, e.g., about the difference between the axioms for a group
and the axioms of ZFC, and intuitionism vs. classical logic.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/24/2005 3:00:36 PM
tchow@lsa.umich.edu wrote in message
<41f50d94$0$563$b45e6eb0@senator-bedfellow.mit.edu>...
>In article <vcbsm4rgvpa.fsf@beta19.sm.ltu.se>,
>Torkel Franzen  <torkel@sm.luth.se> wrote:
>>poopdeville@gmail.com writes:
>>> It's true in every model for the satisfiability relation.
>>  And what does it mean that it's true in every model for the
>>satisfiability relation?
>
>Exactly.  However, I suspect that 'cid 'ooh still doesn't see what I'm
>driving at.  I will try again, although I'm beginning to lose hope.
>
>Recall 'cid 'ooh's objection to my usage of "true": What does it mean
>to say that "the cartesian product of nonempty sets is nonempty" is
>true?  My response was, following Tarski, that I was using "true" in an
>eliminable way, and that by saying "the cartesian product of nonempty
>sets is nonempty" is true, I was saying no more than that the cartesian
>product of nonempty sets is nonempty.  'cid 'ooh balked at this,
>claiming not to understand what it means for the cartesian product of
>nonempty sets to be nonempty unless I specified a model that it was
>true *in*.
>
>My response was that 'cid 'ooh was guilty of exactly the same sin he was
>charging me with, namely asserting mathematical statements flat out without
>saying which model of ZFC they were true *in*.

Right.
The informal use of the notion of truth is governed by Tarski's
disquotational T-scheme, as for example,

   "John Lennon was born in 1940" is true
              if and only if
       John Lennon was born in 1940

After Alfred Tarski described this analysis to Rudolf Carnap (who had been
trying unsuccessfuly to define "true" in various ways), Carnap later
reported that "the scales fell from my eyes".

This is all explained in great detail in any basic textbook which explains
theories of truth (e.g., Richard Kirkham's _Theories of Truth_, Susan
Haack's _Philosophy of Logics_ or W.V. Quine's _Philosophy of Logic_).

Asking which model the sentence "John Lennon was born in 1940" is true in is
a confusion. Its truth value depends upon whether John Lennon was born in
1940. There are interpretations in which "John Lennon was born in 1940" is
false, but this is irrelevant, since we use the term "John Lennon" to refer
to John Lennon, and so on.

The same holds for sentences like "ZFC is consistent" or "There exists an
infinite set", etc. The term "ZFC" refers to formal system ZFC and the
predicate "consistent" refers to sets of sentences such that a contradiction
is not provable. A sentence of the form Fa is true iff the object denoted by
the term a has the property denoted by the predicate F. It follows that

         "ZFC is consistent" is true iff ZFC is consistent.

And hence,
IF ZFC is consistent, then "ZFC is consistent" is true.
IF ZFC is inconsistent, then "ZFC is consistent" is not true.

This has nothing to do with models. It has to do with a correct
understanding (as given by Tarski) of how we use the word "true". Exactly
the same holds for the manner in which Spanish speakers use the word
"verdad", German speakers use "wahr" and French speakers use "vrai".

The tendency to confuse the disquotational analysis of truth (given by
Tarski in 1933) with the notion of "truth-in-a-model" (also given by Tarski
in later papers, and later developing into full-blown model theory) should
be used a measure of a certain kind of intelligence. People who get it are
Tarski-enlightened and people don't get it are Tarski-stupid.

--- Jeff


0
ketland (18)
1/24/2005 3:55:09 PM
tchow@lsa.umich.edu wrote:
> In article <vcbsm4rgvpa.fsf@beta19.sm.ltu.se>,
> Torkel Franzen  <torkel@sm.luth.se> wrote:
> >poopdeville@gmail.com writes:
> >> It's true in every model for the satisfiability relation.
> >  And what does it mean that it's true in every model for the
> >satisfiability relation?
>
> Exactly.  However, I suspect that 'cid 'ooh still doesn't see what
I'm
> driving at.  I will try again, although I'm beginning to lose hope.
>
> Recall 'cid 'ooh's objection to my usage of "true": What does it mean
> to say that "the cartesian product of nonempty sets is nonempty" is
> true?  My response was, following Tarski, that I was using "true" in
an
> eliminable way, and that by saying "the cartesian product of nonempty
> sets is nonempty" is true, I was saying no more than that the
cartesian
> product of nonempty sets is nonempty.  'cid 'ooh balked at this,
> claiming not to understand what it means for the cartesian product of
> nonempty sets to be nonempty unless I specified a model that it was
> true *in*.

I have no quarrel with Tarski's usage (though the redundancy theory of
truth has been discredited for years).  Your usage was slightly
different, however. You *asked* if AC was true.  Your usage and
Tarski's are flatly incompatible.  To wit -- the answer to the question
"Is AC true?" is trivially "No" since we can construct sets in ZF for
which it fails.  I very much doubt this is what you want.

> My response was that 'cid 'ooh was guilty of exactly the same sin he
was
> charging me with, namely asserting mathematical statements flat out
without
> saying which model of ZFC they were true *in*.  As an example, I gave
the
> statement
>
>    (*) ZFC is consistent iff ZFC has models.
>
> I asked, what does this mean?  I don't understand what it means until
you
> tell me which model of ZFC it's true *in*.  (Of course, I do in fact
know
> what it means, but am asking these questions to illustrate my point
that
> 'cid 'ooh's objections can be turned against him.)  'cid 'ooh's
reaction
> was, essentially, that (*) is true in all models of ZFC.

Essentially, my answer doesn't depend on models of ZFC.  It depends on
models of the FO meta-language of ZFC.  As you note a bit later, you
can push me into an infinite regress by asking "In what model is there
a model of the satisfiability relation?"  This supports my claim that
truth is context dependent and not absolute.  I'll admit that I didn't
say "... in model X of the satisfiability relation," but I only omitted
it because I assumed context supported it, as it supports most
mathematical lanugage.  We only run into trouble when we divorce an
utterance from the context it which it occurs.

The sentence "the cartesian product of non-empty sets is non-empty,"
stripped of its context, is virtually meaningless.  Your mind might
wander to thoughts of ZFC or the real numbers, where it's true.  But my
mind might wander to ZF, where it can fail to hold.  If we're, say,
working on a problem together, the context is fixed, we know if we're
working in ZF or ZFC, and we can determine a definite truth value.

>
> But this is not an answer at all.  If it were an answer, then I could
> respond to the question of why I accept that "the cartesian product
of
> nonempty sets is nonempty" by simply observing that it's true in all
> models of ZFC.  That's correct, but is obviously not going to satisfy
> 'cid 'ooh.  Any axiom is true in all models of that axiom; this
trivial
> fact nothing about whether to accept it, and of course is not why I
> accept it.
>

It certainly would satisfy me.  If ZFC is all we're considering, AC is
obviously true.  The question gets more interesting if we consider ZF.
(That is, Zermelo-Fraenkel without choice).  Remember -- what I'm
advocating is context sensitivity, not absolute truth.  A thing can be
true "here" but not "there."

> 'cid 'ooh has a blind spot; when others make mathematical assertions,
> such as "every vector space has a basis," he claims not to be able to
> understand any mathematical statement as having a specific, definite
> meaning unless some model is explicitly specified that it is true
*in*.
> Yet he himself makes mathematical assertions like (*) and like
>
>   (**) Theorems of ZFC, e.g., (*), are true in all models of ZFC
>
> baldly, meaning something specific and definite by them, without
saying
> which model of these statements are true *in*.  When pressed, he says
> that they are true in all models, which obviously is insufficient to
> fix a specific, determinate meaning for the statements in question,
and
> provides no basis for accepting or rejecting (*) or (**)
>
> What 'cid 'ooh needs to acknowledge is that he makes lots of
mathematical
> statements flat out without needing to say which model they're "true
in."
> As Torkel Franzen points out, an example is (**); surely it cannot be
the
> case that (**) makes no sense unless one says which model it's true
in,
> or else we get an infinite regress (which model is "(**) is true in
model X"
> true in?).
>

I thought you were trying to push me into the infinite regress
direction.  Well, there's already an infinite regress in FOL.  In fact,
it's almost the same one:  every first order language requires a model
theory, which needs an underlying set theoretic structure, which in
turn requires a first-order language.  More-or-less, for each language,
you might say you need a new meta-language to understand or "ground"
it.  But each meta-language is a language in itself.  By induction,
kaboom.  This doesn't stop people from using FOL because once you know
what's going in one language/meta-language pair, you know what's going
on in all of them.

Moreover, this sort of infinite regress is to be *expected* from an
infinite sequence of ordered pairs of the form (What does x_{n-1}
mean?, x_n).  Every explanation is a symbol that must be interpreted,
and whose meaning is determined by context.  This line of thought would
take us far afield.  If interested, I suggest Wittgenstein's Blue Book
or Philosophical Investigations.

'cid 'ooh

0
poopdeville (133)
1/24/2005 10:01:11 PM
Now we're getting somewhere.

In article <1106604071.297554.238780@f14g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>I have no quarrel with Tarski's usage (though the redundancy theory of
>truth has been discredited for years).  Your usage was slightly
>different, however. You *asked* if AC was true.  Your usage and
>Tarski's are flatly incompatible.

No, it isn't.  I asked if AC was true.  Following Tarski, this amounts to
asking whether the cartesian product of nonempty sets is nonempty.

>To wit -- the answer to the question
>"Is AC true?" is trivially "No" since we can construct sets in ZF for
>which it fails.  I very much doubt this is what you want.

The fact that there are models of ZF in which AC is false does not settle
one way or the other whether the cartesian product of nonempty sets is
nonempty.

For example, the sentence

   If ZF is consistent then there are models of ZF in which AC is false

is provable in ZF (in fact in much weaker systems) and is consistent with
both AC and its negation.

>The sentence "the cartesian product of non-empty sets is non-empty,"
>stripped of its context, is virtually meaningless.

This is the key claim.  Of course, things have to be taken in context.
The correct context for the sentence in question, however, is the class
of all sets.  It is not ZF or ZFC.

Let's take "ZFC is consistent" for comparison.  It's true in some models
of ZFC and false in other models of ZFC.  Does that mean that it's
meaningless unless I say whether I'm considering the "context" of a
model of ZFC in which it's true or a model in which it's false?  No.
We *know* what it means for ZFC to be consistent, because we understand
what symbols are, what strings are, what rules are, what integers are,
and so forth.  No infinite regress prevents us from grasping these
things.  Furthermore, just because we can exhibit models of ZFC in
which "ZFC is consistent" can go either way doesn't somehow cause us
to lose our ability to understand what we meant by the statement in
the first place.  And we can continue to ask, "Is ZFC consistent?"
(equivalently, is "ZFC is consistent" true?) and know what we're asking
without having to specify any model.  In fact, models are irrelevant.
The correct context for understanding "ZFC is consistent" is the world
of arithmetic, or of syntax (arithmetic and syntax are mutually
intepretable).

Similarly, the fact that AC is true in some models of ZF and false in
others does not imply that it is meaningless for us to ask whether
the cartesian product of nonempty sets is nonempty, in the context of
the class of all sets.

Now, you might be skeptical about sets.  In fact, it's pretty clear
from your discussion that you *do* direct a kind of skepticism towards
(infinite?) sets that you don't direct towards syntactic entities.  I
don't actually have any problem with your choosing to be skeptical of
sets.  What I do object to is the unfounded argument that independence
results, or theories of truth (absolute vs. relative) have anything to
do with such skepticism.  Whether you prefer Tarski or Wittgenstein is
actually irrelevant to the main issue at hand.

By the way, although this is mostly a philosophical debate, I would point
out that I've noticed a rather strong correlation between getting these
"philosophical" issues straight and getting simple mathematical facts
straight.  Recall my simple example above: the sentence

   If ZF is consistent then there are models of ZF in which AC is false

is provable in ZF (in fact in much weaker systems) and is consistent with
both AC and its negation.  This is a straight mathematical fact, acceptable
even to those who are skeptical about sets (but not of syntax).  I suspect,
though, that most people---perhaps 'cid 'ooh included, though maybe I'm
wrong about that---who claim not to understand what "the cartesian product
of nonempty sets is nonempty" means will have some difficulty seeing this
fact and its relevance to the discussion.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/25/2005 3:33:43 PM
poopdeville@gmail.com wrote:
> tchow@lsa.umich.edu wrote:
>>Recall 'cid 'ooh's objection to my usage of "true": What does it mean
>>to say that "the cartesian product of nonempty sets is nonempty" is
>>true?  My response was, following Tarski, that I was using "true" in an
>>eliminable way, and that by saying "the cartesian product of nonempty
>>sets is nonempty" is true, I was saying no more than that the cartesian
>>product of nonempty sets is nonempty.  'cid 'ooh balked at this,
>>claiming not to understand what it means for the cartesian product of
>>nonempty sets to be nonempty unless I specified a model that it was
>>true *in*.
> 
> 
> I have no quarrel with Tarski's usage (though the redundancy theory of
> truth has been discredited for years).  Your usage was slightly
> different, however. You *asked* if AC was true.  Your usage and
> Tarski's are flatly incompatible.  To wit -- the answer to the question
> "Is AC true?" is trivially "No" since we can construct sets in ZF for
> which it fails.  I very much doubt this is what you want.

"Construct sets in ZF for which it fails"?  Hard to figure out
what you mean by this.  There are *models* of ZF that *believe*
it fails.  Let M be such a model, and let x be an element of M
such that M thinks (i) x does not contain the empty set and (ii)
x has an empty Cartesian product.

Now let x^M (the "M-extension of x") be the set of all y in M
such that M thinks y is an element of x.  Then for each y in x^M we
can similarly define y^M, and M is correct that no such y^M is empty.  Now
the question is, is there a function that, to each such y^M,
assigns an element of y^M?  We know that no such function can
be "in" M--that is, there is no f in M such that, if we write f^M
for the function that sends z^M to f(z)^M, for every z that M
thinks is in the domain of f, then f^M has the required property.
It doesn't follow that no such function exists in reality.
0
mike_lists (73)
1/25/2005 4:11:07 PM
In article <35n9cpF4ojl1pU1@individual.net>,
Mike Oliver  <mike_lists@verizon.net> wrote:
>poopdeville@gmail.com wrote:
>> I have no quarrel with Tarski's usage (though the redundancy theory of
>> truth has been discredited for years).  Your usage was slightly
>> different, however. You *asked* if AC was true.  Your usage and
>> Tarski's are flatly incompatible.  To wit -- the answer to the question
>> "Is AC true?" is trivially "No" since we can construct sets in ZF for
>> which it fails.  I very much doubt this is what you want.
>
>"Construct sets in ZF for which it fails"?  Hard to figure out
>what you mean by this.  There are *models* of ZF that *believe* it fails.

Everything you say is of course correct, but unfortunately I'm not sure it
will convey your point.

It occurs to me that there is another tactic.  Given 'cid 'ooh's insistence
on specifying a model in order to determine context, perhaps the simplest
answer is just, "When I ask whether AC is true, I'm asking whether it's
true in V, the class of all sets, which is a [proper class] model of ZFC."

What do you think, 'cid 'ooh?  Does that help you see what I mean by, "Is
AC true?"  I'm asking whether AC is true in V.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/25/2005 4:16:09 PM
Mike Oliver wrote:
> poopdeville@gmail.com wrote:
> > tchow@lsa.umich.edu wrote:
> >>Recall 'cid 'ooh's objection to my usage of "true": What does it
mean
> >>to say that "the cartesian product of nonempty sets is nonempty" is
> >>true?  My response was, following Tarski, that I was using "true"
in an
> >>eliminable way, and that by saying "the cartesian product of
nonempty
> >>sets is nonempty" is true, I was saying no more than that the
cartesian
> >>product of nonempty sets is nonempty.  'cid 'ooh balked at this,
> >>claiming not to understand what it means for the cartesian product
of
> >>nonempty sets to be nonempty unless I specified a model that it was
> >>true *in*.
> >
> >
> > I have no quarrel with Tarski's usage (though the redundancy theory
of
> > truth has been discredited for years).  Your usage was slightly
> > different, however. You *asked* if AC was true.  Your usage and
> > Tarski's are flatly incompatible.  To wit -- the answer to the
question
> > "Is AC true?" is trivially "No" since we can construct sets in ZF
for
> > which it fails.  I very much doubt this is what you want.
>
> "Construct sets in ZF for which it fails"?  Hard to figure out
> what you mean by this.  There are *models* of ZF that *believe*
> it fails.  Let M be such a model, and let x be an element of M
> such that M thinks (i) x does not contain the empty set and (ii)
> x has an empty Cartesian product.
>

"Construct"  was a poor choice of word, since I don't mean it in any
constructivist sense.  "Demonstrate sets in ZF..." would have been
better.

> We know that no such function can
> be "in" M--that is, there is no f in M such that, if we write f^M
> for the function that sends z^M to f(z)^M, for every z that M
> thinks is in the domain of f, then f^M has the required property.
> It doesn't follow that no such function exists in reality.
"In reality"?  What is that supposed to mean?

'cid 'ooh

0
poopdeville (133)
1/25/2005 6:11:49 PM
Mike Oliver wrote:
> poopdeville@gmail.com wrote:
> > tchow@lsa.umich.edu wrote:
> >>Recall 'cid 'ooh's objection to my usage of "true": What does it
mean
> >>to say that "the cartesian product of nonempty sets is nonempty" is
> >>true?  My response was, following Tarski, that I was using "true"
in an
> >>eliminable way, and that by saying "the cartesian product of
nonempty
> >>sets is nonempty" is true, I was saying no more than that the
cartesian
> >>product of nonempty sets is nonempty.  'cid 'ooh balked at this,
> >>claiming not to understand what it means for the cartesian product
of
> >>nonempty sets to be nonempty unless I specified a model that it was
> >>true *in*.
> >
> >
> > I have no quarrel with Tarski's usage (though the redundancy theory
of
> > truth has been discredited for years).  Your usage was slightly
> > different, however. You *asked* if AC was true.  Your usage and
> > Tarski's are flatly incompatible.  To wit -- the answer to the
question
> > "Is AC true?" is trivially "No" since we can construct sets in ZF
for
> > which it fails.  I very much doubt this is what you want.
>
> "Construct sets in ZF for which it fails"?  Hard to figure out
> what you mean by this.  There are *models* of ZF that *believe*
> it fails.  Let M be such a model, and let x be an element of M
> such that M thinks (i) x does not contain the empty set and (ii)
> x has an empty Cartesian product.
>

"Construct"  was a poor choice of word, since I don't mean it in any
constructivist sense.  "Demonstrate sets in ZF..." would have been
better.

> We know that no such function can
> be "in" M--that is, there is no f in M such that, if we write f^M
> for the function that sends z^M to f(z)^M, for every z that M
> thinks is in the domain of f, then f^M has the required property.
> It doesn't follow that no such function exists in reality.
"In reality"?  What is that supposed to mean?

'cid 'ooh

0
poopdeville (133)
1/25/2005 7:10:11 PM
In article <1106676517.056629.39420@c13g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>"Construct"  was a poor choice of word, since I don't mean it in any
>constructivist sense.  "Demonstrate sets in ZF..." would have been
>better.

Mike Oliver's point, or one of his points, is that "sets in ZF" is a
grammatical solecism.  Sets, of the kind you're talking about at least,
exist in a model, not in a system of axioms.

In a sense this is a nitpick, but often this kind of grammatical
mistake is symptomatic of a common conceptual error: that ZF conjures
sets into existence out of thin air.  But if you don't know what sets
are, ZF won't tell you what they are; ZF is just a syntactic object.
Nor does it even really help to mumble something about models, because
what is a model?  The standard definition is that it's a *set*
(equipped with relations and functions corresponding to the relation
symbols and function symbols in your language), and if I don't know
what sets are, then I don't know what a model is either.

>> It doesn't follow that no such function exists in reality.
>"In reality"?  What is that supposed to mean?

In V, the class of all sets.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/25/2005 7:25:36 PM
poopdeville@gmail.com wrote:

> Mike Oliver wrote:

>>"Construct sets in ZF for which it fails"?  Hard to figure out
>>what you mean by this.  There are *models* of ZF that *believe*
>>it fails.  Let M be such a model, and let x be an element of M
>>such that M thinks (i) x does not contain the empty set and (ii)
>>x has an empty Cartesian product.
>>
> 
> 
> "Construct"  was a poor choice of word, since I don't mean it in any
> constructivist sense.  "Demonstrate sets in ZF..." would have been
> better.

I don't follow that either.  What does it mean to demonstrate
a set in ZF?  I hate to sound like Torkel; I just really don't
know what you're getting at here.

>>We know that no such function can
>>be "in" M--that is, there is no f in M such that, if we write f^M
>>for the function that sends z^M to f(z)^M, for every z that M
>>thinks is in the domain of f, then f^M has the required property.
>>It doesn't follow that no such function exists in reality.
> 
> "In reality"?  What is that supposed to mean?

Is there a real object, independent of our reasoning about it,
that satisfies the (now-erased, but hopefully you remember) required
property?  You know, in Plato's heaven, in the mind of God, in
some previously unsuspected physical form made of unobtanium and
lying on a planet a non-Archimedean distance from Earth?  Doesn't
really matter where or when; the consequences are the same.
0
mike_lists (73)
1/25/2005 10:50:51 PM
Mike Oliver wrote:
> poopdeville@gmail.com wrote:
>
> > Mike Oliver wrote:
>
> >>"Construct sets in ZF for which it fails"?  Hard to figure out
> >>what you mean by this.  There are *models* of ZF that *believe*
> >>it fails.  Let M be such a model, and let x be an element of M
> >>such that M thinks (i) x does not contain the empty set and (ii)
> >>x has an empty Cartesian product.
> >>
> >
> >
> > "Construct"  was a poor choice of word, since I don't mean it in
any
> > constructivist sense.  "Demonstrate sets in ZF..." would have been
> > better.
>
> I don't follow that either.  What does it mean to demonstrate
> a set in ZF?  I hate to sound like Torkel; I just really don't
> know what you're getting at here.

It's lazily phrased and not particularly important, but I'll get to it
below.  (OK, another attempt at re-wording:  "There demonstrably exists
models of ZF (whose elements are sets, but I suspect this will be
contentious) in which AC fails.")

>
> >>We know that no such function can
> >>be "in" M--that is, there is no f in M such that, if we write f^M
> >>for the function that sends z^M to f(z)^M, for every z that M
> >>thinks is in the domain of f, then f^M has the required property.
> >>It doesn't follow that no such function exists in reality.
> >
> > "In reality"?  What is that supposed to mean?
>
> Is there a real object, independent of our reasoning about it,
> that satisfies the (now-erased, but hopefully you remember) required
> property?  You know, in Plato's heaven, in the mind of God, in
> some previously unsuspected physical form made of unobtanium and
> lying on a planet a non-Archimedean distance from Earth?  Doesn't
> really matter where or when; the consequences are the same.

I see.  Let's just consider plain old models of ZF and ZFC for a
moment.  Inserting the missing property, the question becomes "Is there
a real object, independent of our reasoning about it, that satisfies
AC?"  I suppose the answer to that question depends on whether or not
the objects in models of ZFC are real.  Now, suppose they are.  How
does this make the elements of a model M for which ZF but not AC hold
unreal?

A realist might say that they are real, but that they aren't the sorts
of things we mean by "set."  However, in my (limited) experience,
mathematicians don't care about the objects they're proving things
about, but the provable relationships between these objects -- after
all, a number theorist doesn't care if the latest craze in modelling PA
uses von Neumann ordinals, atoms of unobtainium (I like that.  I just
got my hands on some unobtainium. :-), or Julius Caesars.

Of course, this isn't a convincing argument that these things in M
*are* sets.  But it does raise the issue of making ad hoc ontological
commitments that fly in the face of mathematical usage.  If the objects
in M are sets, then according to Frege's disquotational scheme (1916,
beating Tarski by 17 years!), AC is false even though it holds for some
sets.

This, and its immediate consequence: context sensitivity, are the
reasons why I prefer my analysis of truth.  It *should* satisfy
realists since it offers some indexicality with which to differentiate
between elements of M and sets in the sense I foisted on you, and as a
direct consequence accomodates different colloquial uses of the term
"set," which I find attractive.

'cid 'ooh

0
poopdeville (133)
1/26/2005 12:42:15 AM
tchow@lsa.umich.edu wrote:
> In article <35n9cpF4ojl1pU1@individual.net>,
> Mike Oliver  <mike_lists@verizon.net> wrote:
> >poopdeville@gmail.com wrote:
> >> I have no quarrel with Tarski's usage (though the redundancy
theory of
> >> truth has been discredited for years).  Your usage was slightly
> >> different, however. You *asked* if AC was true.  Your usage and
> >> Tarski's are flatly incompatible.  To wit -- the answer to the
question
> >> "Is AC true?" is trivially "No" since we can construct sets in ZF
for
> >> which it fails.  I very much doubt this is what you want.
> >
> >"Construct sets in ZF for which it fails"?  Hard to figure out
> >what you mean by this.  There are *models* of ZF that *believe* it
fails.
>
> Everything you say is of course correct, but unfortunately I'm not
sure it
> will convey your point.
>
> It occurs to me that there is another tactic.  Given 'cid 'ooh's
insistence
> on specifying a model in order to determine context, perhaps the
simplest
> answer is just, "When I ask whether AC is true, I'm asking whether
it's
> true in V, the class of all sets, which is a [proper class] model of
ZFC."
>
> What do you think, 'cid 'ooh?  Does that help you see what I mean by,
"Is
> AC true?"  I'm asking whether AC is true in V.

Of course that works for me.  But is that what you really meant before?
If it is, then why didn't you just say so?  Not to be a dick, but this
explanation looks like it's meant just to get me to shut up.  If you
don't want to play, that's fine with me -- we can just agree to
disagree.  

(By the way, V |= AC.  :-)

'cid 'ooh

0
poopdeville (133)
1/26/2005 12:58:38 AM
poopdeville@gmail.com wrote:
> Mike Oliver wrote:
> It's lazily phrased and not particularly important, but I'll get to it
> below.  (OK, another attempt at re-wording:  "There demonstrably exists
> models of ZF (whose elements are sets, but I suspect this will be
> contentious) in which AC fails.")

This looks promising; I may be able to get my point across just from
here.

Let's take a model of ZF whose elements are uncontroversially
sets, in which AC fails.  The model I have in mind is L(R).  It's
a proper class model, but that's easily gotten around if you like
(at the cost of weakening ZF slightly, or strengthening the metatheory
slightly).  L(R) is the class of all sets that you get by starting
with R at the bottom, and then doing the L construction.  L(R)
is a *transitive* model, so the epsilon relation means exactly what
it does out in V--that saves us from doing all these translations
("M-extensions") that I had to deal with in a previous post.

Now (assuming large cardinals, but again we can get around that if
you want to), there is no wellorder of R in L(R), so L(R) satisfies
~AC.  (Equivalently, there is no choice function in L(R) for P(R)\{0}.)

But that doesn't mean there really *isn't* a wellorder of R!  It just
never showed up in L(R), which is missing some sets.  Specifically,
any relation on R, including a wellorder, can be coded by a set
of reals (work this out for yourself), so L(R) is missing any set
of reals that codes a wellorder of R.

Now, the statement "AC is true" just means that this is what's
always going to happen--whenever you have a model whose elements
are sets, in which some set cannot be wellordered, it simply
means that the relevant wellorder is not an element of the
model.

I believe this should take care of all the points you raised in the
rest of your post, so I'm snipping.  Let me know if you feel it
doesn't answer some of them.
0
mike_lists (73)
1/26/2005 2:43:35 AM
poopdeville@gmail.com wrote in message

>If the objects
>in M are sets, then according to Frege's disquotational scheme (1916,
>beating Tarski by 17 years!), AC is false even though it holds for some
>sets.

You mean Frege's "Der Gedanke"? This appeared in 1918-9, not 1916.
Tarski presented his work in 1930-1, although the Polish book did not appear
until 1933, and the German translation, "Der Wahrheitsbegriff" not until
1935-6.
There is also circumstantial evidence that Goedel had developed a certain
amount of formalized semantics too, by 1930-1 (when he independently
obtained both Tarski's indefinability theorem and the result that enriching
a system with higher types permits an inductive truth definition: this
yields a proof of consistency and thus a non-conservative extension).

In any case, Frege had already stated the propositional form of the
equivalence scheme 26 years before 1918, in 1892:

    One can indeed say: "The thought that 5 is a prime number
    is true". But closer examination shows that nothing more has
    been said than in the simple sentence "5 is a prime number".
    (Frege 1892, "Ueber Sinn und Bedeutung", p. 34; in Geach
    and Black 1980, p. 62; and Beaney 1997, p. 158.)

--- Jeff


0
ketland (18)
1/26/2005 3:16:46 AM
In article <1106701118.837557.94530@z14g2000cwz.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>>  I'm asking whether AC is true in V.
>
>Of course that works for me.  But is that what you really meant before?

Sure.

>If it is, then why didn't you just say so?

All the things I said were, following standard mathematical usage,
equivalent to this.  It was not clear to me exactly where your deviations
from standard mathematical usage lay, so I had to fish around with various
attempts to probe where exactly your boundaries were.

>Not to be a dick, but this
>explanation looks like it's meant just to get me to shut up.

Frankly, I'm quite surprised that this gets you to shut up.  For example:

>(By the way, V |= AC.  :-)

I'm not sure how to interpret your ":-)" but I have to ask, how do you know
that V |= AC ?  This is a genuine question, not a rhetorical one like some
of the other ones I've asked.  I'm genuinely surprised that you balk so hard
at the question "Is AC true?" and have no problems at all with "Does V |= AC?"
What distinction are you drawing between them?  In standard mathematical
talk they're identical.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/26/2005 3:20:06 AM
tchow@lsa.umich.edu wrote:
> In article <1106701118.837557.94530@z14g2000cwz.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> >>  I'm asking whether AC is true in V.
> >
> >Of course that works for me.  But is that what you really meant
before?
>
> Sure.
>
> >If it is, then why didn't you just say so?
>
> All the things I said were, following standard mathematical usage,
> equivalent to this.  It was not clear to me exactly where your
deviations
> from standard mathematical usage lay, so I had to fish around with
various
> attempts to probe where exactly your boundaries were.
>
> >Not to be a dick, but this
> >explanation looks like it's meant just to get me to shut up.
>
> Frankly, I'm quite surprised that this gets you to shut up.  For
example:
>
> >(By the way, V |= AC.  :-)
>
> I'm not sure how to interpret your ":-)" but I have to ask, how do
you know
> that V |= AC ?  This is a genuine question, not a rhetorical one like
some
> of the other ones I've asked.

Well, given that V is a model for ZFC -- Zermelo-Fraenkel with Choice
-- V |= AC is the fact that V satisfies AC.

> I'm genuinely surprised that you balk so hard
> at the question "Is AC true?" and have no problems at all with "Does
V |= AC?"
> What distinction are you drawing between them?  In standard
mathematical
> talk they're identical.

Your (and Mike's) explanation gave me exactly what I wanted -- truth
relative to a particular model.  I don't particularly *like* that this
is the standard usage, since it gives ontological priority to V. But
just as no one would preface every English sentence with "In the real
numbers, ..." when doing real analysis, nobody would preface every
sentence with "In V, ..." when doing set theory. Context is context and
convention is convention. I'll learn from our discussion and live with
it.

Now that we've got this misunderstanding out of the way, I wonder what
you think of my points about context and why I don't think this is a
good usage of "true."  Basically, paraphrasing and re-organizing my
comments from before:  (Open) Sentences are true of the things that
satisfy them.  Models capture this relation.  Saying that a particular
model determines the Truth of a sentence even if other models disagree
with it is ad hoc.

'cid 'ooh

0
poopdeville (133)
1/26/2005 7:19:22 AM
poopdeville@gmail.com says...

>>...how do you know that V |= AC?...
>Well, given that V is a model for ZFC -- Zermelo-Fraenkel with Choice
>-- V |= AC is the fact that V satisfies AC.

V is not defined to be a model for ZFC. It is defined to be the
class 

    { x | exists an ordinal alpha such that x is in V_alpha }

where the sets V_alpha are defined recursively as follows:

    1. if alpha = 0, then V_alpha = the empty set
    2. if alpha = beta + 1 for some ordinal beta, then V_alpha = the
    set of all subsets of V_beta
    3. if alpha is a limit ordinal, then V_alpha = the union of all V_beta
    for beta < alpha.

It doesn't follow from the definition of V that it is a model of ZFC.
It only follows that it is a model of ZF.

--
Daryl McCullough
Ithaca, NY

0
1/26/2005 1:51:40 PM
tchow@lsa.umich.edu wrote:
> In article <35n9cpF4ojl1pU1@individual.net>,
> Mike Oliver  <mike_lists@verizon.net> wrote:
> 
>>poopdeville@gmail.com wrote:
>>
>>>...the answer to the question
>>>"Is AC true?" is trivially "No" since we can construct sets in ZF for
>>>which it fails.  I very much doubt this is what you want.
>>
>>"Construct sets in ZF for which it fails"?  Hard to figure out
>>what you mean by this.  

I was stumped by that one as well. My guess was that he meant to say "in ZF 
+ not AC". I don't quite see how either interpretation helps his argument.

>>There are *models* of ZF that *believe* it fails.

What does it mean for a model to "think" or "believe"? I can guess, but it 
seems an odd usage to me. Is it common?

> ... unfortunately I'm not sure it will convey your point.

That is certainly true.

> It occurs to me that there is another tactic.  Given 'cid 'ooh's insistence
> on specifying a model in order to determine context, perhaps the simplest
> answer is just, "When I ask whether AC is true, I'm asking whether it's
> true in V, the class of all sets, which is a [proper class] model of ZFC."

By V do you mean some *particular* model? I think you do, but have no way 
to tell exactly which one. For example, I don't know if V is a model in 
which CH is true or not.

I suppose "the class of all sets" might be seen as specifying V, but I 
don't see how it will help in a discussion with someone who is skeptical of 
set theory.

In any case, AC is a theorem of ZFC, so it is true in any model of ZFC. So 
any ambiguity is not material.

> Does that help you see what I mean by, "Is
> AC true?"  I'm asking whether AC is true in V.

But earlier in this thread when I said:
> Ralph Hartley <hart...@aic.nrl.navy.mil> wrote:
>>when mathematicians make unqualified statements, with no other
>>context, they usually *mean* "In  ZFC".
You said:
> As a matter of sociological fact, this is definitely false.

The only difference I can see between my interpretation and yours is that, 
you refer to a particular model V.

Do different mathematicians mean different things by V, so that some mean a 
model in which CH is true, and some a model in which it is false? Such 
ambiguity would be harmless when discussing statements that don't depend on 
CH etc.

Or is it your position that there is a single "intended" model V and if 
people disagree about its properties, some are right and some are wrong?

Does it make any practical difference which of those positions we accept? I 
seriously doubt it.

Ralph Hartley
0
hartley (156)
1/26/2005 2:27:04 PM
In article <1106723962.638685.9760@f14g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Well, given that V is a model for ZFC -- Zermelo-Fraenkel with Choice
>-- V |= AC is the fact that V satisfies AC.

But how do we know that V is a model for ZFC?  V is just the class of
all sets.  People only say that V is a model of ZFC because they
believe that the cartesian product of a nonempty family of nonempty
sets is nonempty.  If you didn't believe that, you also wouldn't
believe that V is a model of ZFC.  So I don't see how bringing V into
the picture solves your problem of determining whether V |= AC, given
that you claim not to know what it means for the cartesian product of
a nonempty family of nonempty sets to be nonempty, let alone whether
it's true.

>Now that we've got this misunderstanding out of the way, I wonder what
>you think of my points about context and why I don't think this is a
>good usage of "true."  Basically, paraphrasing and re-organizing my
>comments from before:  (Open) Sentences are true of the things that
>satisfy them.  Models capture this relation.  Saying that a particular
>model determines the Truth of a sentence even if other models disagree
>with it is ad hoc.

First of all, the way mathematicians *in fact* use the word "true"
violates your norms.  They may be guilty of a philosophical transgression,
but if you want to understand what they're saying, you have to get used
to this usage whether you like it or not.  I think this is how the whole
(sub)thread went flailing off in the first place: You just weren't used
to normal mathematical talk.

That aside, the more interesting question is whether mathematicians are
guilty of abusing language, or of implicitly adhering to a "discredited"
theory of truth, or of some other philosophical sin.

The major problem I see with your insistence that truth makes sense only
relative to a model is that the infinite regress that we briefly discussed
earlier becomes vicious.  I assert a sentence S.  You claim not to
understand what S means unless I say which model it's true *in*.  So I
assert the sentence S' = "S is true in model M."  But there's no reason
to stop here: Why not challenge S' as meaningless unless you say which
model it's true in?  Then when I try to assert S'' = "S' is true in M'"
....well, it's obvious where this leads.

The only way I see of breaking out of this regress is to simply assert
by fiat that we know what certain sentences mean.  We know what symbols
are, what rules are, what integers are, and so forth, so we know what
we mean by "ZFC is consistent" simpliciter.  We might not know whether
ZFC *is* in fact consistent, but we know what it would *mean* for it to
be consistent.

In a sense this "assertion by fiat" is arbitrary.  I'm arbitrarily saying
that I know, for example, that I know what it means for a proof to have
"finite" length, and that I know what the standard integers are and that
they're not the nonstandard integers.  However, this is the only way that
I see to get off the ground, and symbols, rules, integers, etc., are about
the simplest mathematical objects around; if we can't assume that we know
what *they* are, then we're basically taking the defeatist position that
doing mathematics is impossible.  Therefore I am not bothered by the
arbitrariness.

The advantage of this point of view is that we can then develop mathematics
as usual.  Once in a while we might get into factional arguments; some
people might feel that infinite sets are perfectly clear while others
might not, and some might like the law of the excluded middle and others
might not.  The general outlines, however, are the same.  We have some
statements that we make, and we know what they mean.  *Then* you can go
about developing mathematical logic, and analyzing mathematical discourse
by mimicking it with formal languages.

One then discovers that if I mimic mathematical discourse with a formal
language, then that formal language admits different interpretations.
This should not be a surprise; *of course* I can redefine words and
sentences to mean anything I want them to.  In particular, I don't see
why I should suddenly lose faith in my original "arbitrary" meaning of
"ZFC is consistent" and doubt that I knew what I meant in the first
place, just because someone else can make "ZFC is consistent" mean
something totally different.

What I've just said is typically convincing to many people, but those
same people often balk at, say, AC or the continuum hypothesis.  Then
the independence results seem to make people want to say that AC or CH
is "meaningless" (or some such) unless you specify a model.  However,
I maintain that this is just because these folks didn't feel they had
a clear idea of what sets were in the first place.  Independence results
by themselves don't necessarily force this upon you, as the case of
"ZFC is consistent" should make clear.

Here's another way to put it.  There are certain informal mathematical
statements that we make, that we know the meaning of---e.g., "ZFC is
consistent."  We study this informal discourse by mimicking it with
formal discourse---let me use "Con(ZFC)" to indicate a formal sentence
that mimics "ZFC is consistent."  "Con(ZFC)" by itself is indeed
meaningless; I have to specify a model before it has a meaning and
before I can begin to evaluate its truth.  But this doesn't mean that
"ZFC is consistent" is meaningless or ambiguous.

The case of group theory is different.  We can use the same formal
apparatus, but now we're not mimicking general mathematical discourse.
In general mathematical discourse we don't say "For all a, b, and c,
(a*b)*c = a*(b*c)" and feel that we know exactly what we're saying
without any explanation of what a, b, and c are or what * is.  So
then it doesn't make sense to ask whether the statement is "true"
without further explication.  But this doesn't undermine the previous
discussion, because this is a different ball game.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/26/2005 7:13:28 PM
In article <ct8928$ijm$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>What does it mean for a model to "think" or "believe"? I can guess, but it 
>seems an odd usage to me. Is it common?

M thinks that X if X is true in M.  It's not really a formal technical term;
it's just an informal locution that is often helpful when trying to keep
track of what's going on in a complicated situation.  It's fairly common,
but only when people are being very informal.

>By V do you mean some *particular* model? I think you do, but have no way 
>to tell exactly which one. For example, I don't know if V is a model in 
>which CH is true or not.
>
>I suppose "the class of all sets" might be seen as specifying V, but I 
>don't see how it will help in a discussion with someone who is skeptical of 
>set theory.

I do mean some particular model, namely the class of all sets, and I agree
that I wasn't sure it would help in this discussion.  But if you've been
following the rest of this thread, you'll see that (to my surprise) it did.

>> Ralph Hartley <hart...@aic.nrl.navy.mil> wrote:
>>>when mathematicians make unqualified statements, with no other
>>>context, they usually *mean* "In  ZFC".
>You said:
>> As a matter of sociological fact, this is definitely false.
>
>The only difference I can see between my interpretation and yours is that, 
>you refer to a particular model V.

Well, for starters, there is in fact a big difference between ZFC and V.
ZFC is just one system of axioms for sets.  If we discarded it, we
could replace it with some other axioms for set theory, but V would
still be around until we decided to abandon Cantor's paradise because
we didn't like infinite sets any more.

More to the point, though, for many mathematical statements, such as
S = "every differentiable function is continuous," it's not even clear
that mathematicians really mean "S is true in V" when they assert S.
Mathematicians feel that they understand what differentiable functions
and continuity are, and although there is a small amount of set theory
buried in the basic definitions, the full glory of V is not necessarily
implicitly present.  For example, people have shown that a lot of
analysis can be developed with reference just to integers and sets of
integers (any maybe sets of sets of integers).  So when they assert S,
they might only be thinking of S as interpreted in this smaller universe.
There are a lot of options, and unless being precise about the details
of your mathematical universe is important (usually it isn't), people
usually leave these unspecified.  The simplest way, I think, of describing
this situation is that mathematicians mean S when they say S.  If you
press the question, "But in what universe?" the answer will be "any
suitable universe, sufficient for what I'm studying right now."

This isn't true for all mathematical statements.  In particular, for
something like AC, I think that when people assert AC, then they mean
that AC is true in V, because AC is inherently a sweeping set-theoretical
statement about all sets.

>Do different mathematicians mean different things by V, so that some mean a 
>model in which CH is true, and some a model in which it is false? Such 
>ambiguity would be harmless when discussing statements that don't depend on 
>CH etc.
>
>Or is it your position that there is a single "intended" model V and if 
>people disagree about its properties, some are right and some are wrong?

The latter.

>Does it make any practical difference which of those positions we accept? I 
>seriously doubt it.

Well, I'm not sure how you'd flesh out the concept that "different
mathematicians mean different things by V."  So I can't tell if it would
make a practical difference.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/26/2005 7:53:12 PM
Daryl McCullough wrote:

> V is not defined to be a model for ZFC. It is defined to be the
> class 
> 
>     { x | exists an ordinal alpha such that x is in V_alpha }
> 
> where the sets V_alpha are defined recursively as follows:
> 
>     1. if alpha = 0, then V_alpha = the empty set
>     2. if alpha = beta + 1 for some ordinal beta, then V_alpha = the
>     set of all subsets of V_beta
>     3. if alpha is a limit ordinal, then V_alpha = the union of all V_beta
>     for beta < alpha.
> 
> It doesn't follow from the definition of V that it is a model of ZFC.
> It only follows that it is a model of ZF.

I think you might have to explain what you mean by that, Daryl.

In some sense, all truths about V follow from the definition,
by its categoricity:  Up to a unique isomorphism there's only
one object that answers to the definition.  (Well, strictly
speaking there's none--V doesn't exist as a completed totality--
but for each alpha, there's up to a unique isomorphism only
one V_alpha.)

Presumably that's not what you meant, unless you're asserting
that you believe AC is really false.  But what did you mean?
In what sense does, say, Replacement follow from the definition,
that Choice doesn't?
0
mike_lists (73)
1/26/2005 10:14:13 PM
Mike Oliver says...
>
>Daryl McCullough wrote:
>
>> V is not defined to be a model for ZFC. It is defined to be the
>> class 
>> 
>>     { x | exists an ordinal alpha such that x is in V_alpha }
....
>> It doesn't follow from the definition of V that it is a model of ZFC.
>> It only follows that it is a model of ZF.
>
>I think you might have to explain what you mean by that, Daryl.
>
>In some sense, all truths about V follow from the definition,
>by its categoricity:  Up to a unique isomorphism there's only
>one object that answers to the definition.

That's what I mean. It isn't that V is defined to be a model of
ZFC; its definition is the same whether you are working in Z, ZF,
ZFC, ZFC+GCH, or whatever. It's just that what you can *prove* about
V is different in these different theories.

--
Daryl McCullough
Ithaca, NY

0
1/26/2005 11:00:58 PM
Daryl McCullough wrote:
> Mike Oliver says...
>>Daryl McCullough wrote:
>>>It doesn't follow from the definition of V that it is a model of ZFC.
>>>It only follows that it is a model of ZF.
>>
>>I think you might have to explain what you mean by that, Daryl.
>>
>>In some sense, all truths about V follow from the definition,
>>by its categoricity:  Up to a unique isomorphism there's only
>>one object that answers to the definition.
> 
> 
> That's what I mean. It isn't that V is defined to be a model of
> ZFC; its definition is the same whether you are working in Z, ZF,
> ZFC, ZFC+GCH, or whatever. It's just that what you can *prove* about
> V is different in these different theories.

Sorry, I still don't see what you're gettng at when you
say it follows from the definition that it's a model of ZF, but
not that it's a model of ZFC.  Are you saying nothing more
than that's what follows *from*ZF* ?  I thought you had
some deeper point in mind.
0
mike_lists (73)
1/27/2005 12:23:17 AM
In article <35qqjhF4pn616U1@individual.net>,
Mike Oliver  <mike_lists@verizon.net> wrote:
>Sorry, I still don't see what you're gettng at when you
>say it follows from the definition that it's a model of ZF, but
>not that it's a model of ZFC.

I was wondering about that too.

At the risk of derailing the conversation somewhat, let me ask something
that I've been wondering about.  When I say "V," I generally mean just the
class of all sets.  However, do most set theorists mean something more?
I'm thinking of some of Kunen's informal discussion in his book on
set theory.  One of the first restrictions he makes has to do with the
hereditary property: A cow is not a set, so we typically don't want to
consider the "set of all cows" to be a set.  Then when discussing the
axiom of foundation, in the form "V = WF," he says that there's no need
to regard this as saying that all sets are "really" well-founded but
only that we're restricting our attention to the cumulative hierarchy.
Finally, of course, Kunen takes the formal definition of V to be the
formula "x = x" since he doesn't want to get into the question of what
a proper class is.

This makes me think that Kunen thinks of V as having the hereditary
property more or less "by definition," but that V isn't the cumulative
hierarchy "by definition" but only out of convenience.  So in particular
he wouldn't define V as Daryl has.

Is Kunen's point of view typical, or idiosyncratic?  Or perhaps (like many
things in mathematics) it's a matter of taste and context how precise we
are about what V is?
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/27/2005 1:27:57 AM
Mike Oliver says...

>Sorry, I still don't see what you're gettng at when you
>say it follows from the definition that it's a model of ZF, but
>not that it's a model of ZFC.  Are you saying nothing more
>than that's what follows *from*ZF* ?  I thought you had
>some deeper point in mind.

No, I didn't.

--
Daryl McCullough
Ithaca, NY

0
daryl5382 (108)
1/27/2005 4:15:25 AM
tchow@lsa.umich.edu says...

>At the risk of derailing the conversation somewhat, let me ask something
>that I've been wondering about.  When I say "V," I generally mean just the
>class of all sets.  However, do most set theorists mean something more?

....

>Then when discussing the
>axiom of foundation, in the form "V = WF," he says that there's no need
>to regard this as saying that all sets are "really" well-founded but
>only that we're restricting our attention to the cumulative hierarchy.
>Finally, of course, Kunen takes the formal definition of V to be the
>formula "x = x" since he doesn't want to get into the question of what
>a proper class is.
>
>This makes me think that Kunen thinks of V as having the hereditary
>property more or less "by definition,"

What is the hereditary property?

>but that V isn't the cumulative
>hierarchy "by definition" but only out of convenience.

>So in particular
>he wouldn't define V as Daryl has.

Well, there are several different definitions that are provably
equivalent if we assume foundation (and replacement):

   1. U (the universe) = { x | x=x }
   2. WF (the well-founded sets) = { x | x is well-founded }
   3. V (the cumulative hierarchy) 
      = { x | exists alpha such that x is an element of V_alpha }

But in a theory such as Aczel's set theory (or Quine's NF, maybe),
you can regard V as a definable subclass of U. I think WF and V
have to be the same, as long as replacement is true.

--
Daryl McCullough
Ithaca, NY

0
1/27/2005 4:29:07 AM
Our posts are quickly becoming very long.  I apologize for the lack of
snipping.


tchow@lsa.umich.edu wrote:
> In article <1106723962.638685.9760@f14g2000cwb.googlegroups.com>,
>  <poopdeville@gmail.com> wrote:
> >Well, given that V is a model for ZFC -- Zermelo-Fraenkel with
Choice
> >-- V |= AC is the fact that V satisfies AC.
>
> But how do we know that V is a model for ZFC?  V is just the class of
> all sets.  People only say that V is a model of ZFC because they
> believe that the cartesian product of a nonempty family of nonempty
> sets is nonempty.  If you didn't believe that, you also wouldn't
> believe that V is a model of ZFC.

Well, you claimed that V was a model for ZFC when you wrote:

"When I ask whether AC is true, I'm asking whether it's
true in V, the class of all sets, which is a [proper class] model of
ZFC."

I just assumed you had some technical knowledge I didn't, since you've
obviously studied this much more than I have.

<snip>

> First of all, the way mathematicians *in fact* use the word "true"
> violates your norms.  They may be guilty of a philosophical
transgression,
> but if you want to understand what they're saying, you have to get
used
> to this usage whether you like it or not.  I think this is how the
whole
> (sub)thread went flailing off in the first place: You just weren't
used
> to normal mathematical talk.
>

Of course.  I addressed this in the paragraph you snipped.  However,
normal mathematical talk is still quite confused.  Here I refer to your
(and Jeffrey Ketland's, but mostly his) talk of disquotational schemes,
with emphasis of Jeffrey's use of models and the real world and your
group theory example at the end of your message.

> That aside, the more interesting question is whether mathematicians
are
> guilty of abusing language, or of implicitly adhering to a
"discredited"
> theory of truth, or of some other philosophical sin.
>

I think mathematicians make outrageous ontological claims when doing
philosophy.  But in most cases the result of their mathematical work is
going to be independent of their philosophical biases.

> The major problem I see with your insistence that truth makes sense
only
> relative to a model is that the infinite regress that we briefly
discussed
> earlier becomes vicious.  I assert a sentence S.  You claim not to
> understand what S means unless I say which model it's true *in*.  So
I
> assert the sentence S' = "S is true in model M."  But there's no
reason
> to stop here: Why not challenge S' as meaningless unless you say
which
> model it's true in?  Then when I try to assert S'' = "S' is true in
M'"
> ...well, it's obvious where this leads.

It's not vicious if you use new models to provide a foundation for the
old ones.  The way I see it (philosophically, anyways) is that a model
is something like a linguistic explanation of meaning.

Consider the following example -- it's relevance will hopefully become
clear:  You're walking on a forest trail, but it suddenly ends, right
in the middle of the forest.  However, you see a sign with an arrow on
it:  <-.  Presumably, if you go in the correct direction, you'll find
the trail again.  How are you justified in going left?  Of course,
convention dictates that left is the correct direction.  But suppose
you need to explain what the glyph "<-" means.  Would you say "That
means left" or point in that direction?  "<-" is just a symbol and must
be interpreted.  And your explanation is too.  If a symbol is
misinterpreted, and if one still wishes to explain what is meant, one
must provide a new symbol.  (By induction, kaboom)  Explanation piles
upon explanation until (hopefully) the listener understands.  But he
might never understand.  Does that mean that explanations are useless?

Clearly not.  Just because there *might* be an infinite regress doesn't
mean that there *is* one.  Explanations are very often successful in
communicating what we intend.  The analogy here is that an explanation
provides a foundation for what it attempts to explain, just as each
term in my (possibly transfinite) sequence of models provides a
foundation for the previous one.  Once you get what I'm driving at,
there's no need for any more.

> The only way I see of breaking out of this regress is to simply
assert
> by fiat that we know what certain sentences mean.  We know what
symbols
> are, what rules are, what integers are, and so forth, so we know what
> we mean by "ZFC is consistent" simpliciter.  We might not know
whether
> ZFC *is* in fact consistent, but we know what it would *mean* for it
to
> be consistent.
>
> In a sense this "assertion by fiat" is arbitrary.  I'm arbitrarily
saying
> that I know, for example, that I know what it means for a proof to
have
> "finite" length, and that I know what the standard integers are and
that
> they're not the nonstandard integers.  However, this is the only way
that
> I see to get off the ground, and symbols, rules, integers, etc., are
about
> the simplest mathematical objects around; if we can't assume that we
know
> what *they* are, then we're basically taking the defeatist position
that
> doing mathematics is impossible.  Therefore I am not bothered by the
> arbitrariness.

You don't need to know what numbers are to do number theory.  Or sets
for set theory.  Frege struggled for years trying to prove that Julius
Caesar wasn't a number.  But this wasn't for any mathematical reason,
just a philosophical one.  I'm skeptical of your (and my) knowledge of
what a number is, but I don't claim we can't reason about them.

> The advantage of this point of view is that we can then develop
mathematics
> as usual.  Once in a while we might get into factional arguments;
some
> people might feel that infinite sets are perfectly clear while others
> might not, and some might like the law of the excluded middle and
others
> might not.  The general outlines, however, are the same.  We have
some
> statements that we make, and we know what they mean.  *Then* you can
go
> about developing mathematical logic, and analyzing mathematical
discourse
> by mimicking it with formal languages.
>

My view is also that mathematicians drive mathematics.  However, the
existence of these factional differences implies that not all
mathematicians share the same intuitions about the sorts of objects
they're dealing with.  In order to communicate their results with other
mathematicians, one has to give them enough information to evaluate the
truth of their claims.  In intuitionist analysis, every function is
continuous.  This is spectacularly false in classical analysis.  But a
classical mathematician must accept the proof that intuitionist
analysis implies that every function is continuous since every step in
the proof is valid in classical logic.  They're just playing different
games.  If they want to play with one another, they need to communicate
their rules.

> One then discovers that if I mimic mathematical discourse with a
formal
> language, then that formal language admits different interpretations.
> This should not be a surprise; *of course* I can redefine words and
> sentences to mean anything I want them to.

Agreed.

> In particular, I don't see
> why I should suddenly lose faith in my original "arbitrary" meaning
of
> "ZFC is consistent" and doubt that I knew what I meant in the first
> place, just because someone else can make "ZFC is consistent" mean
> something totally different.

Because if you can't communicate your arbitrary meaning of "ZFC is
consistent," the phrase really is meaningless.  Whereas what can be
proven (and is thus true in all relevant models) is meaningful.  As is
a statement if evaluated with respect to a particular interpretation.

>
> What I've just said is typically convincing to many people, but those
> same people often balk at, say, AC or the continuum hypothesis.  Then
> the independence results seem to make people want to say that AC or
CH
> is "meaningless" (or some such) unless you specify a model.  However,
> I maintain that this is just because these folks didn't feel they had
> a clear idea of what sets were in the first place.  Independence
results
> by themselves don't necessarily force this upon you, as the case of
> "ZFC is consistent" should make clear.

Perhaps.  I have no issue *using* AC, but I certainly fall into the
categories you describe (assuming that "(or some such) is a modulo
relation)  But what makes you so sure *you* know what a set is if you
can't even communicate it?

<snip>

> The case of group theory is different.  We can use the same formal
> apparatus, but now we're not mimicking general mathematical
discourse.
> In general mathematical discourse we don't say "For all a, b, and c,
> (a*b)*c = a*(b*c)" and feel that we know exactly what we're saying
> without any explanation of what a, b, and c are or what * is.  So
> then it doesn't make sense to ask whether the statement is "true"
> without further explication.  But this doesn't undermine the previous
> discussion, because this is a different ball game.

I still feel that this division is artificial.  A set of axioms for FOL
mimics the discourse of a particular field.  But Intuitionism
demonstrates that even FOL + arbitrary axioms isn't enough to mimic the
discourse of an arbitrary field.  In short, there is no "general
mathematical discourse," only instances of mathematical discourse.
Also note that by this standard it doesn't make sense to say that "AC
is true" without explication of what is meant by "true" -- truth
relative to V.

'cid 'ooh

0
poopdeville (133)
1/27/2005 11:06:24 AM
In article <ct9qmj0101e@drn.newsguy.com>,
Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
>What is the hereditary property?

Any member of a set is a set.

>Well, there are several different definitions that are provably
>equivalent if we assume foundation (and replacement):

Right.  However, Kunen hedges somewhat about whether V = WF is "really"
true (at least at first; after "clearing his throat" he then states that
he's going to take V = WF as a basic axiom without further comment).
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/27/2005 2:17:46 PM
poopdeville@gmail.com wrote in message
>tchow@lsa.umich.edu wrote:
>> First of all, the way mathematicians *in fact* use the word "true"
>> violates your norms.  They may be guilty of a philosophical
>transgression,
>> but if you want to understand what they're saying, you have to get
>used
>> to this usage whether you like it or not.  I think this is how the
>whole
>> (sub)thread went flailing off in the first place: You just weren't
>used
>> to normal mathematical talk.
>>
>
>Of course.  I addressed this in the paragraph you snipped.  However,
>normal mathematical talk is still quite confused.  Here I refer to your
>(and Jeffrey Ketland's, but mostly his) talk of disquotational schemes,
>with emphasis of Jeffrey's use of models and the real world and your
>group theory example at the end of your message.

Sorry. I've snipped the rest, but this really is the heart of the matter.
Mathematical talk is *not confused*. Rather, you seem to be advocating some
unmotivated, and possibly incoherent, form of scepticism.

Consider Goldbach's Conjecture, GC. Its truth condition is stated as
follows:
GC is *true* if and only if for every even number n, there are primes p1 and
p2 such that n = p1 + p2.

At present, we do not know if GC is true or false. But this is a precise
analysis of what saying "GC is true" means.

The above is an instance of the partial defintion giving what the word
"true" means. It illustrates how the word "true" is actually used, both in
ordinary life, in science and in mathematics. Truth for interpreted
statements is intrinsically disquotational, just as 7 is intrinsically
prime. This has nothing to do with the "redundancy theory", which Tarski
refuted. It is a central property of the notion of truth.
(Proof: Let (L, I) be an interpreted language and let T be the set of truths
in (L, I). Suppose that dom(I) contains all the expressions of L. Suppose
that L also contains a predicate True(x) which defines this set T. For each
expression E, let E* be a term in L such that E* denotes (in I) E. Then, for
any A in L, each sentence True(A*) <-> A is true in (L, I). Disquotation is
an intrinsic property of truth.
More generally, if (ML, MI) is a meta-language for (L, I), and True(x) is a
formula in ML which defines truth in (L, I)) and there is a translation t
which maps L-expressions to ML-expressions, then we get that True(A*) <->
t(A) is true in (ML, MI), for each sentence A of L. This is Tarski's
Convention T.)

We have a precise and exact mathematical theory of truth (for arithmetic).
The set of arithmetic truths is precisely defined. Suppose we take L_{PA}
with just ~, & and "forall" as primitive logical expressions.

   Tr(N) is the smallest set X such that
   (a) X is a subset of Sent(L_{PA})
   (b) for any closed terms t, u, t=u is in X iff val(t) = val(u)
   (c) for any A in Sent(L_{PA}), ~A is in X iff A is not in X
   (d) for any A, B in Sent(L_{PA}), A&B is in X iff A is in X and B is in X
   (e) for any A, for any v, "forall v,A" is in X iff, for any n in N,
A(n/v) is in X

This set is Sigma^1_1. It is not definable in arithmetic. Etc.
This body of mathematical work began in the 1930's with Alfred Tarski and
has developed a great deal since (Mostowski, Feferman, Kripke, Friedman,
etc.). If you have a precise objection to this serious and correct analysis
of truth, then what is your objection?

Of course, one can also talk of truth for *uninterpreted formulas*, using
the notion of truth-in-an-interpretation (NOT model---this assumes you have
some axioms around), also defined by Alfred Tarski. This concerns
uninterpreted formulas and structures, and whether such a formula is true
relative to that structure. Thus,
      Fab is true in I if and only if (a^I, b^I) is in F^I.
Note that semantic theory itself is riddled with set theory. F^I is, for
example, a subset of the Cartesian product D^2 where D is the domain (i.e.,
a set) of I.

But GC is not an uninterpreted formula. It is a meaningful statement about
the numbers.
Your argument requires that we literally identify a meaningful statement and
its logical form (the associated uninterpreted formula), thereby denying
that mathematical statements are meaningful statements.
I can think of no good reason for doing this. Not even nominalists do this
(they simply deny the existence of numbers, sets, etc., tout court).
To illustrate, let GC be Goldbach and let GC* be its logical form. Then we
have:

GC: For any even number n>2, there are primes p and q such that n = p + q

GC*:  Ax(Fx & Rxa -> EyEz(Gy & Gz & x = f(y,z))

Similarly, from baby logic, we have things like:

S : Lennon is taller than McCartney
S*: Pab

By the way, that it isn't hard to prove that
    GC is true if and only if N |= GC*
Similarly, in the semantical meta-theory for the language of ZF, we can
prove
     AC is true if and only if V |= AC*
as well as
     AC is true if and only if every set of the right sort has a choice set.

If you want to argue that mathematical statements are meaningless (and thus
should be identified with their uninterpreted logical forms), then you
really need to give a precise argument for this radically sceptical claim.
I see no relevant difference between GC and S. Whether GC is true depends
upon the properties of even numbers and primes; whether S is true depends
upon the properties of John and Paul.

Furthermore, if you think that ordinary mathematical statements are
meaningless, but also that meta-mathematical statements about models are
meaningful, then you seem to be contradicting yourself, as Tim pointed out.
Indeed, what is a model but a set?

(A set-sized structure for the language of ZF is a pair (D, R), where D is a
non-empty set, and R is a subset of the set D^2. A structure (D, R) is a
model of ZF if and only if all axioms of ZF are true in (D, R).
This is why one cannot intelligibly define "set" in terms of "model of ZF".
It is incoherently circular. In contrast, one can define "group" in terms of
"model of group axioms G1, G2, G3". Thus, (D, o) is a group iff D is
non-empty set and o is a binary associative operation on D, with a unit,
unique inverse, etc.)

Also, how exactly would you define "A is true in M" without a theory of
sequences, etc.?
Come to think of it, if you're advocating some sort of radical scepticism
about meaning, what is the "intended interpretation" for this post to
sci.logic?
E.g., why are statements about numbers meaningless, but, say, "Jeff was born
in England" meaningful?
Everyone agrees that semantical theory is difficult (e.g., in natural
languages there is ambiguity, vagueness, indexicality, intensionality,
etc.), but what you are saying about the semantics of mathematical
statements doesn't make much sense. For another example, on your account, we
cannot even deal with minimal *applications* of mathematics to the physical
world, as in "The number of elephants in London Zoo is exactly 5" or "The
axial-vector function that represents the magnetic field has zero
divergence".

--- Jeff



0
ketland (18)
1/27/2005 2:47:57 PM
In article <ct8928$ijm$1@ra.nrl.navy.mil>,
Ralph Hartley  <hartley@aic.nrl.navy.mil> wrote:
>But earlier in this thread when I said:
>> Ralph Hartley <hart...@aic.nrl.navy.mil> wrote:
>>>when mathematicians make unqualified statements, with no other
>>>context, they usually *mean* "In  ZFC".
>You said:
>> As a matter of sociological fact, this is definitely false.
>
>The only difference I can see between my interpretation and yours is that, 
>you refer to a particular model V.

I feel that I didn't answer this adequately in my other article, so here's
some more comment.

The first point is that ZFC is a set of axioms whereas V is the class of all
sets, so they're fundamentally different kinds of entities, and so "in ZFC"
and "in V" are fundamentally different kinds of predicates.  I can make a
claim of the form

    [1]  "S" really means "S is true in V"

but to say

    [2]  "S" really means "S is true in ZFC"

is a grammatical blunder.  When you said "in ZFC" I think you probably meant
something like

    [3]  "S" really means "S is provable in ZFC"

Now, [1] is somewhat controversial, but at least there's plenty of room for
debate on both sides.  As I explained in my other article, I think [1] makes
sense for S = AC, but is less plausible for your average mathematical
statement that makes use of only very low levels of the cumulative
hierarchy.

On the other hand, [3] is much less tenable.  For example, if [3] were true,
then you would expect that

    [4]  S  iff  S is provable in ZFC

would be provable in a very weak theory, for any S.  But this isn't the
case.  For instance, if CH denotes the continuum hypothesis, then

    [5]  CH  ->  CH is provable in ZFC

isn't provable even in ZFC (unless ZFC+Con(ZFC) is inconsistent), let alone
a weak theory.  Conversely, if Con(ZFC) denotes "ZFC is consistent," then

    [6]  Con(ZFC) is provable in ZFC  ->  Con(ZFC)

isn't provable in ZFC unless ZFC is inconsistent.  (For Goedel's 2nd theorem
is provable in ZFC, so

    [7]  Con(ZFC) is provable in ZFC  ->  ~ Con(ZFC)

is provable in ZFC.  So if [6] were also provable in ZFC, then ZFC would
disprove "Con(ZFC) is provable in ZFC," i.e., ZFC would prove that something
is not provable in ZFC, so in particular ZFC would prove that ZFC is
consistent, which implies that ZFC is inconsistent by Goedel's 2nd theorem
again.)  In fact, [6] is an instance of something called a "reflection
principle."  It's very plausible informally, because if we "believe in"
ZFC then we believe that anything provable in ZFC is actually true.
However, one consequence of Goedel's results is that this plausible
principle is, logically speaking, an additional assumption over and above
the axioms of ZFC themselves.

Another way to remember that "S" and "S is provable in ZFC" are very
different statements is to think about what it takes to formalize each
of these statements.  To formalize "S is provable in ZFC" we need to
formalize the concept of provability, which is a nontrivial exercise.
The provability predicate is a monstrously complicated thing if written
out in full gory detail.  So "S is provable in ZFC" is necessarily a
monstrously long formal sentence, regardless of S.  In contrast,
formalizing "S" itself could be easy if "S" is a simple statement.

What is probably generating confusion is the familiar sociological fact
that mathematicians do not confidently assert mathematical statements
unless they have a proof of that statement available.  If someone tells
you confidently, "Every differentiable function is continuous," you can
rightly infer that (provided he is a trustworthy person) he knows a proof
(or at least knows *of* a proof) that every differentiable function is
continuous.  But the fact that I won't assert S unless I know how to
justify S doesn't mean that "S" *means* "S is justifiable."  This holds
not only in mathematics but in ordinary discourse.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/27/2005 3:02:12 PM
In article <1106823984.872929.137690@z14g2000cwz.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Well, you claimed that V was a model for ZFC when you wrote:
>"When I ask whether AC is true, I'm asking whether it's
>true in V, the class of all sets, which is a [proper class] model of
>ZFC."

My bad.  Since I do in fact believe that AC is true, I easily slip into
saying things that presuppose its truth.  But you're right that if I'm
posing the question "Is AC true?" in what's supposed to be a meaningful
way, then what I said is garbled.  If I had been more careful, I would
just have said:

  When I ask whether AC is true, I'm asking whether it's
  true in V, the class of all sets.

I shouldn't have said that V is a model of ZFC in this context, even
though that is commonly assumed in mathematics.

>Explanations are very often successful in
>communicating what we intend.  The analogy here is that an explanation
>provides a foundation for what it attempts to explain, just as each
>term in my (possibly transfinite) sequence of models provides a
>foundation for the previous one.  Once you get what I'm driving at,
>there's no need for any more.

What I find curious about your account here is that I largely agree with it,
but draw different conclusions than you do from it.  You don't think the
regress is infinite because in practice it bottoms out somewhere.  I agree,
and describe the situation by saying that at the bottoming-out point, I
grasp your meaning *without* needing to ask for truth-in-a-model.  I grasp
your meaning simpliciter.  I don't see why you don't describe the situation
the same way.

So for example, I assert AC.  You ask, "AC in which model?"  I take this to
be a symptom of the fact that you are skeptical about sets, so you can't
grasp AC simpliciter.  I've bottomed out at AC; you're bottoming out
somewhere else.  But either way, at some point we know what's meant without
having to say "In what model?"  So there's some other kind of notion of
truth/meaning coming into play there.

>You don't need to know what numbers are to do number theory.  Or sets
>for set theory.  Frege struggled for years trying to prove that Julius
>Caesar wasn't a number.  But this wasn't for any mathematical reason,
>just a philosophical one.  I'm skeptical of your (and my) knowledge of
>what a number is, but I don't claim we can't reason about them.

But don't you at least need to know what symbols are, and what syntactic
rules are?  Are you skeptical about rules?  I suppose if you like
Wittgenstein, maybe you are.  But then how can you do any mathematics
if you don't know what symbols, strings, and rules are?

>Because if you can't communicate your arbitrary meaning of "ZFC is
>consistent," the phrase really is meaningless.

I agree with that.  But in practice, there is no trouble communicating
this arbitrary meaning except to extreme skeptics.  And we shouldn't
expect to be able to communicate with extreme skeptics.

>Whereas what can be
>proven (and is thus true in all relevant models) is meaningful.

Is it?  You have to be able to communicate your proof.  What if you can't
do that?  USENET provides spectacularly good examples of how even the most
perspicuously transparent proofs fail to be accepted by everyone.

>But what makes you so sure *you* know what a set is if you
>can't even communicate it?

If indeed I couldn't communicate it, I would be worried.  But I don't have
any trouble in practice communicating it.  I experience no communication
difficulties when I talk with professional set theorists, logicians, and
mathematicians.  It's only when talking with students (and on USENET!)
that communication difficulties arise, and there the difficulties arise
not because of lack of clarity of the notion of a set, but for other
reasons.

>I still feel that this division is artificial.  A set of axioms for FOL
>mimics the discourse of a particular field.

The set of axioms for a group does not mimic the *discourse* of group
theorists.  Group theorists will assert things like, "Every odd order
group is solvable."  Or "Every finite simple group is either a cyclic
group of prime order, an alternating group, a group of Lie type, or one
of 26 sporadic exceptions."  Or "There exist infinite groups that are
finitely generated with finite exponent."  None of these statements can
be expressed in the first-order language of group theory.  Only certain
properties of groups---first-order properties---can be expressed in
this language.  The function of the group axioms is to define what a
group *is*, and not to mimic *discourse* about groups.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/27/2005 4:42:01 PM
Jeffrey Ketland wrote:
> poopdeville@gmail.com wrote in message
> >tchow@lsa.umich.edu wrote:
> >> First of all, the way mathematicians *in fact* use the word "true"
> >> violates your norms.  They may be guilty of a philosophical
> >transgression,
> >> but if you want to understand what they're saying, you have to get
> >used
> >> to this usage whether you like it or not.  I think this is how the
> >whole
> >> (sub)thread went flailing off in the first place: You just weren't
> >used
> >> to normal mathematical talk.
> >>
> >
> >Of course.  I addressed this in the paragraph you snipped.  However,
> >normal mathematical talk is still quite confused.  Here I refer to
your
> >(and Jeffrey Ketland's, but mostly his) talk of disquotational
schemes,
> >with emphasis of Jeffrey's use of models and the real world and your
> >group theory example at the end of your message.
>
> Sorry. I've snipped the rest, but this really is the heart of the
matter.
> Mathematical talk is *not confused*. Rather, you seem to be
advocating some
> unmotivated, and possibly incoherent, form of scepticism.

What is incoherent is claimng that the truth conditions for the
sentence "'AC' is true" are the same as the truth conditions for "'John
Lennon was born in 1940' is true" with the explanation that the second
is true because "John Lennon" refers to a real object.  What is the
real object to which "AC" refers?  You explicitly chosen "the real
world" to be the set of objects with respect to which the second
sentence is to be evaluated.  But you implicitly chose V (or some other
structure) with respect to which evaluate the first and failed to
supply this information, leaving the reader to suppose that you mean
"the real world."

My motivations are my own.

>
> Consider Goldbach's Conjecture, GC. Its truth condition is stated as
> follows:
> GC is *true* if and only if for every even number n, there are primes
p1 and
> p2 such that n = p1 + p2.
>
> At present, we do not know if GC is true or false. But this is a
precise
> analysis of what saying "GC is true" means.
>
> The above is an instance of the partial defintion giving what the
word
> "true" means. It illustrates how the word "true" is actually used,
both in
> ordinary life, in science and in mathematics. Truth for interpreted
> statements is intrinsically disquotational, just as 7 is
intrinsically
> prime. This has nothing to do with the "redundancy theory", which
Tarski
> refuted. It is a central property of the notion of truth.

You're doing it again.  Where do these numbers live?  Certainly not the
same place John Lennon does.  By the way, the disquotational theory of
truth and the redundacy theory of truth are synonymous.  This is basic
in philosophy, but this case that I'm guilty of obfuscation too.

> (Proof: Let (L, I) be an interpreted language and let T be the set of
truths
> in (L, I). Suppose that dom(I) contains all the expressions of L.
Suppose
> that L also contains a predicate True(x) which defines this set T.
For each
> expression E, let E* be a term in L such that E* denotes (in I) E.
Then, for
> any A in L, each sentence True(A*) <-> A is true in (L, I).
Disquotation is
> an intrinsic property of truth.

A ha!   You're actually using the standard formulation of a model
theory to describe truth (via the disquotational-theory-esque
construction that a sentence S is true iff I(S) = T, where I is the
interpretation function).  This is how I would proceed.  But if you
don't fix I, you have no basis with which to interpret.  This is
EXACTLY my point.

'cid 'ooh

0
poopdeville (133)
1/28/2005 1:22:11 AM
tchow@lsa.umich.edu wrote:
> In article <1106823984.872929.137690@z14g2000cwz.googlegroups.com>,
><snip>
> >Explanations are very often successful in
> >communicating what we intend.  The analogy here is that an
explanation
> >provides a foundation for what it attempts to explain, just as each
> >term in my (possibly transfinite) sequence of models provides a
> >foundation for the previous one.  Once you get what I'm driving at,
> >there's no need for any more.
>
> What I find curious about your account here is that I largely agree
with it,
> but draw different conclusions than you do from it.  You don't think
the
> regress is infinite because in practice it bottoms out somewhere.  I
agree,
> and describe the situation by saying that at the bottoming-out point,
I
> grasp your meaning *without* needing to ask for truth-in-a-model.  I
grasp
> your meaning simpliciter.  I don't see why you don't describe the
situation
> the same way.
>
> So for example, I assert AC.  You ask, "AC in which model?"  I take
this to
> be a symptom of the fact that you are skeptical about sets, so you
can't
> grasp AC simpliciter.  I've bottomed out at AC; you're bottoming out
> somewhere else.  But either way, at some point we know what's meant
without
> having to say "In what model?"  So there's some other kind of notion
of
> truth/meaning coming into play there.

Now that we have a common language to work with, many of my claims are
uncontroversial.  However, we've both brought some philosophical
baggage to the table.  In my understanding of your position, you're
committed to the existence of sets.

>
> >You don't need to know what numbers are to do number theory.  Or
sets
> >for set theory.  Frege struggled for years trying to prove that
Julius
> >Caesar wasn't a number.  But this wasn't for any mathematical
reason,
> >just a philosophical one.  I'm skeptical of your (and my) knowledge
of
> >what a number is, but I don't claim we can't reason about them.
>
> But don't you at least need to know what symbols are, and what
syntactic
> rules are?  Are you skeptical about rules?  I suppose if you like
> Wittgenstein, maybe you are.  But then how can you do any mathematics
> if you don't know what symbols, strings, and rules are?

Yes!  We need to know what symbols are, and what (the relevant)
syntactic rules are to do number theory.  But the syntactic rules
demonstrably don't pick out a unique interpretation, which is exactly
why we need to be careful when talking about truth.

<snip>
> >Whereas what can be
> >proven (and is thus true in all relevant models) is meaningful.
>
> Is it?  You have to be able to communicate your proof.  What if you
can't
> do that?  USENET provides spectacularly good examples of how even the
most
> perspicuously transparent proofs fail to be accepted by everyone.
>

This is an interesting point.  I'll have to think about it some.

> >But what makes you so sure *you* know what a set is if you
> >can't even communicate it?
>
> If indeed I couldn't communicate it, I would be worried.  But I don't
have
> any trouble in practice communicating it.  I experience no
communication
> difficulties when I talk with professional set theorists, logicians,
and
> mathematicians.  It's only when talking with students (and on
USENET!)
> that communication difficulties arise, and there the difficulties
arise
> not because of lack of clarity of the notion of a set, but for other
> reasons.
>

I was perhaps dishonestly attributing to you an inability to
communicate what a set is.  If you look in just about any book on set
theory, it'll say that sets are collections of objects or something
just as unsatisfactory.  Pinter's "Set Theory" even states:

"Every axiomatic system, as we have seen, must start with a certain
number of undefined notions.... While we are free in our own minds to
attach a "meaning," in the form of a mental picture, to each of these
notions, mathematically we must proceed "as if" we did not know what
they meant.  Now an "undefined" notion has no properties except those
which are explicitly assigned to it; therefore, we must state as axioms
all the elementary properties we expect our undefined notions to have."

If you can tell me what a set is, outside of a "collection" (the
more-or-less personal "mental picture") or "an element of a collection
which satisfies the membership relation" (the public notion that
doesn't pick out one of these mental pictures), that'd be great.  :-)

> >I still feel that this division is artificial.  A set of axioms for
FOL
> >mimics the discourse of a particular field.
>
> The set of axioms for a group does not mimic the *discourse* of group
> theorists.  Group theorists will assert things like, "Every odd order
> group is solvable."  Or "Every finite simple group is either a cyclic
> group of prime order, an alternating group, a group of Lie type, or
one
> of 26 sporadic exceptions."  Or "There exist infinite groups that are
> finitely generated with finite exponent."  None of these statements
can
> be expressed in the first-order language of group theory.  Only
certain
> properties of groups---first-order properties---can be expressed in
> this language.  The function of the group axioms is to define what a
> group *is*, and not to mimic *discourse* about groups.
I had not thought of this distinction.  

'cid 'ooh

0
poopdeville (133)
1/28/2005 2:01:01 AM
poopdeville@gmail.com wrote in message
>What is incoherent is claimng that the truth conditions for the
>sentence "'AC' is true" are the same as the truth conditions for "'John
>Lennon was born in 1940' is true" with the explanation that the second
>is true because "John Lennon" refers to a real object.

I think you mean "truth conditions for the sentence AC" and "truth
conditions for "John Lennon was born in 1940" ".
Never mind. I didn't say that "the second is true because "John Lennon"
refers to a real object". What I stated was the T-sentence, expressing its
truth condition,

    "John Lennon was born in 1940" is true
           if and only if
     John Lennon was born in 1940.

I have not said that the statement mentioned is true. It might be *false*.
If John Lennon was born in 1939, and Aunt Mimi fibbed, then the statement is
false. But it doesn't matter.
By the way, the T-scheme itself is very *weak*: it doesn't imply direct
assertions of the form "A is true". More precisely, the
[non-self-applicative] T-scheme can be added conservatively to any
sufficiently rich theory. See J. Ketland 1999, "Deflationism and Tarski's
Paradise", Mind 108, pp. 69-94. Or V. Halbach 1999, "Conservative Theories
of Classical Truth", Studia Logica.
The T-scheme simply expresses the truth conditions of (interpreted)
statements.

>What is the
>real object to which "AC" refers?

I don't know what a "real object" is. I tend to think that John Lennon is
less real than the number 2, but don't tell anyone I said that.
What I do know is that AC is the statement "Every set of disjoint non-empty
sets has a choice set". This quantifies over sets. What do you think it
quantifies over?

If I get your main view, it's this: AC is an uninterpreted formula. I.e.,
you think that AC is the uninterpreted formula

  (Aa)[(Ax)(Ay)((Rxa & Rya & x =/= y) => ~(Ew)Rwx&Rwy)
  & (Ax)(Rxa => EwRwa)] => (Ec)(Ax)(Rxa => (Eu)(Aw)(Rwc & Rwx <=> w = u))

Whereas Tim, Mike and I think that AC is the meaningful statement

     Every set of non-empty disjoint sets has a choice set.

Things would be a lot easier if you could just clear this up.

Connectedly, if I get what you're hinting at, you seem to be advocating some
sort of nominalism: sets aren't "real objects"? If so, why not simply say
so? Then we can drop all the discusion of *truth*? After all,
nominalists---people who don't believe in numbers, sets and so on---all
*accept* that AC is true iff every set has a choice set. I know a few
prominent nominalists. They accept the truth condition of AC, but deny that
there are any sets. None of this has anything to do with truth. (Nominalists
tend to be disquotationalists about truth, but that's another matter.)

>You explicitly chosen "the real
>world" to be the set of objects with respect to which the second
>sentence is to be evaluated.

Set of objects? I thought you didn't believe in sets!

>But you implicitly chose V (or some other
>structure) with respect to which evaluate the first and failed to
>supply this information, leaving the reader to suppose that you mean
>"the real world."

When I write messages in English to sci.logic I actually mean Jane Austen's
world. You're meant to go and read every work in the Jane Austen canon, and
find out if the relevant sentence is there. If it's there, then it's "true".
If its negation is there, then it's "false". If neither, then it lacks a
truth value.
Just kidding, just kidding.

>> Consider Goldbach's Conjecture, GC. Its truth condition is stated as
>> follows:
>> GC is *true* if and only if for every even number n, there are primes
>p1 and
>> p2 such that n = p1 + p2.
>>
>> At present, we do not know if GC is true or false. But this is a
>precise
>> analysis of what saying "GC is true" means.
>>
>> The above is an instance of the partial defintion giving what the
>word
>> "true" means. It illustrates how the word "true" is actually used,
>both in
>> ordinary life, in science and in mathematics. Truth for interpreted
>> statements is intrinsically disquotational, just as 7 is
>intrinsically
>> prime. This has nothing to do with the "redundancy theory", which
>Tarski
>> refuted. It is a central property of the notion of truth.
>
>You're doing it again.  Where do these numbers live?

How could a number "live somewhere".
It's like asking if democracy is the square root of the Eiffel Tower.

Numbers are either
(i) abstract entities of some sort (the realist view)
(ii) mental constructions of some sort (constructivist view)
(iii) or there aren't any at all (nominalism).

But this is all irrelevant. The (correct) T-sentence,

   GC is true iff for any even n > 2, there are primes p, q such that n =
p+q

has exactly nothing to do with where numbers "live", or even whether there
are any numbers. Like I said before, it's conservative.

>Certainly not the
>same place John Lennon does.

OK. Progress. We have that:
(a) John Lennon is (was) a physical object of some sort, and
(b) Numbers are not physical objects.
So, in the end, are you simply claiming that there aren't any non-physical
objects? In other words, are you trying to defend nominalism (or formalism)?
If so, then your ontology consists of physical objects, and this will
include concrete expression tokens. And not, by the way, models, which are
abstract entities; also, not expression types, such as arbitrarily long
formulas; or proofs, which are arbitrarily long sequences of abstract
expression types (formulas).

Let's suppose that you want to be a nominalist. OK. If so, what does this
have to do with *truth*? Nothing, so far as I can tell. If there aren't any
non-physical objects, then the usual statements of mathematics ("there is an
empty set", "there is an infinite set") are simply *false*. Indeed, this is
what my nominalist friends tell me.

>By the way, the disquotational theory of
>truth and the redundacy theory of truth are synonymous.

They're not. The redundancy theory as usually understood was advocated by
Alfred Ayer (in 1936) and it is sometimes alleged that it was advocated by
Frank Ramsey in his "Facts and Propositions" paper, but this is somewhat
controversial, since Ramsey recognized a problem with generalizations
expressed using the truth predicate.

In contrast, the rather different disquotational theory of truth is
advocated by Hartry Field, with precedents including Stephen Leeds and
perhaps some scattered remarks of Quine. The disquotational view is
described in several important papers by Hartry Field, anthologised in his
_Truth and the Absence of Fact_ (2001). It is also advocated by Volker
Halbach in some recent publications.
Different to both is the minimalist view which has been defended by Paul
Horwich (e.g., _Truth_, 1990, and several recent publications).
And there's a performative version of deflationism associated with Strawson,
and sometimes Rorty makes similar noises.
And there's the anaphoric view of Grover.

These views all come under the general rubric: deflationary theories of
truth. For nice discussions, see either
(A) the introduction by Simon Blackburn and Keith Simmons to their 1999
anthology _Truth_ (OUP)
(B) the introduction to the section on deflationary theories by Michael
Lynch in his 2001 anthology _The Nature of Truth_ (MIT).
(C) the introduction by Volker Halbach and Leon Horsten to their 2002 volume
_Principles of Truth_.)

>This is basic
>in philosophy.

No it isn't. It would lead to a poor mark in a term essay.

>> (Proof: Let (L, I) be an interpreted language and let T be the set of
>truths
>> in (L, I). Suppose that dom(I) contains all the expressions of L.
>Suppose
>> that L also contains a predicate True(x) which defines this set T.
>For each
>> expression E, let E* be a term in L such that E* denotes (in I) E.
>Then, for
>> any A in L, each sentence True(A*) <-> A is true in (L, I).
>Disquotation is
>> an intrinsic property of truth.
>
>A ha!   You're actually using the standard formulation of a model
>theory to describe truth (via the disquotational-theory-esque
>construction that a sentence S is true iff I(S) = T, where I is the
>interpretation function).  This is how I would proceed.

This may continue forever....
There's an intuitive difference between:

(a) truth for *interpreted* statements (Tarski 1933: "The Concept of Truth
in Formalized Languages")
(b) truth-in-a-structure for *uninterpreted* formulas (Tarski 1935: "On the
Concept of Logical Consequence").

It is the notion discussed in (a) that has the disquotational property.
When we speak, the statements we exchange are *interpreted* statements ("Hi,
I live in Scotand. My dog has no nose."), and the correct truth predicate is
(to a first approximation) disquotational. The second approximation concerns
vagueness, ambiguity, indexicality and intensional operators. This is how
real people actually use the truth predicate in their reasoning. There's
very little dispute about this.

Now, there is an interpretation somehow attached or built-in to the
interpreted statements we use. But no one knows how that works. It's a major
problem---maybe the central problem---in semantical theory. David Lewis and
Scott Soames (and Kripke in seminar talks) have suggested that an
interpreted language is an abstract object. That's what I've called (L, I).
The problem is: what is it about a speaker's usage of L-expressions that
makes her a speaker of interpreted language (L, I) rather than (L, I*)? What
is it about my use of expressions that makes me a speaker of English (where
"cat" refers to cats) rather than Quinglish (where "cat" refers to the
mereological complements of cats)?

Other people (of the more verificationist and Wittgensteinian tradition)
think that it involves an internal grasp of rules for reasoning with and
using (in particular, verifying and refuting) statements.

>But if you
>don't fix I, you have no basis with which to interpret.  This is
>EXACTLY my point.

Right.
But your central point is this: AC is an uninterpreted formula. And you
haven't given an argument for this sceptical claim (except maybe that you
don't know where sets "live"). I don't know where sets "live" either, but
that doesn't make me a radical meaning-sceptic.
Curiously, however, you also think that meta-statements about models are
interpreted and assertible. Indeed, you assert them!
It is these two assumptions which are incoherent (i.e., sentences containing
"set" are uninterpreted; sentences containing "model" are interpreted).

To reiterate Tim's point, consider the interpreted (meta-)statement about
models:
   If ZF has a model, then ZF has a model in which AC is false.
Why isn't this an uninterpreted formula?

Also, you think there is some fundamental difference in the *semantics*
between the statements:

          John Lennon was born in Scotland
          For any even n >2, there are primes p, q such that n = p+q.

It's true that John Lennon is (was) a physical object and 2 is an abstract
object, but this is an *ontological* distinction, not a semantical
distinction. For semantics, the situation is the same. Actually it *has* to
be the same. Otherwise mathematics would not be applicable, a point which
goes back to Frege's Grundgesetze, Vol i. (For an important discussion of
this issue, that a similar kind of truth-conditional semantical theory
applies to mathematical statements as to non-mathematical statements, see
Benacerraf 1973, "Mathematical Truth", in Putnam & Benacerraf (eds.) 1983,
_Philosophy of Mathematics_ (CUP).)
There is nothing "incoherent" about this. In fact, it appears to be
necessary if mathematics is to be applicable.

For example, do you think that

    (*)     The number of elephants in London Zoo is exactly 5

is an uninterpreted formula or an interpreted statement? Elephants are
physical (we know where they "live") but 5 is abstract (we don't know where
it "lives"). And what of London Zoo? Are places or human functional
institutions physical or abstract? What is the mass of London Zoo?
In any case, (*) is a mixed statement, the sort of statement necessary to
any application of mathematics. But, on your view (as I understand it), (*)
is meaningless. By the way, consistent nominalists (like Field, Azzouni,
Leng) think that (*) is false, and would be false even if "5" were replaced
by any other numeral.

--- Jeff


0
ketland (18)
1/28/2005 4:37:14 AM
poopdeville@gmail.com wrote in message
>Yes!  We need to know what symbols are, and what (the relevant)
>syntactic rules are to do number theory.  But the syntactic rules
>demonstrably don't pick out a unique interpretation, which is exactly
>why we need to be careful when talking about truth.

Which "syntactic rules" do you have in mind?
Why do you believe that "syntactic rules" should "pick out a unique
interpretation"?
How exactly it is "demonstrated" that syntactic rules "don't pick out a
unique interpretation"? What assumptions will you use in your
"demonstration"? Are your assumptions true? What rules of reasoning will you
use? Are they truth-preserving?
What is an interpretation? A set with a certain assignment function? What is
a set? What is a function?
How do you define "A is true-in-interpretation I"? How do you define
"formula"?
If "syntactic rules demonstrably don't pick out a unique interpretation",
then can you explain why this implies "we need to be careful when talking
about truth"? (It is consistent with the premise that the interpretation is
"picked out" some other way. How do you know?)

(It's easy to pretend to be a sceptic.)

--- Jeff


0
ketland (18)
1/28/2005 5:15:00 AM
Jeffrey Ketland wrote:
> poopdeville@gmail.com wrote in message
> >What is incoherent is claimng that the truth conditions for the
> >sentence "'AC' is true" are the same as the truth conditions for
"'John
> >Lennon was born in 1940' is true" with the explanation that the
second
> >is true because "John Lennon" refers to a real object.
>
> I think you mean "truth conditions for the sentence AC" and "truth
> conditions for "John Lennon was born in 1940" ".
> Never mind. I didn't say that "the second is true because "John
Lennon"
> refers to a real object". What I stated was the T-sentence,
expressing its
> truth condition,

You wrote:
Asking which model the sentence "John Lennon was born in 1940" is true
in is
a confusion. Its truth value depends upon whether John Lennon was born
in
1940. There are interpretations in which "John Lennon was born in 1940"
is
false, but this is irrelevant, since we use the term "John Lennon" to
refer
to John Lennon, and so on.

>
>     "John Lennon was born in 1940" is true
>            if and only if
>      John Lennon was born in 1940.
>
<snip>
>
> >What is the
> >real object to which "AC" refers?
>
> I don't know what a "real object" is. I tend to think that John
Lennon is
> less real than the number 2, but don't tell anyone I said that.
> What I do know is that AC is the statement "Every set of disjoint
non-empty
> sets has a choice set". This quantifies over sets. What do you think
it
> quantifies over?
>
> If I get your main view, it's this: AC is an uninterpreted formula.
I.e.,
> you think that AC is the uninterpreted formula
>
>   (Aa)[(Ax)(Ay)((Rxa & Rya & x =/= y) => ~(Ew)Rwx&Rwy)
>   & (Ax)(Rxa => EwRwa)] => (Ec)(Ax)(Rxa => (Eu)(Aw)(Rwc & Rwx <=> w =
u))
>
> Whereas Tim, Mike and I think that AC is the meaningful statement
>
>      Every set of non-empty disjoint sets has a choice set.
>
> Things would be a lot easier if you could just clear this up.
>
> Connectedly, if I get what you're hinting at, you seem to be
advocating some
> sort of nominalism: sets aren't "real objects"? If so, why not simply
say
> so? Then we can drop all the discusion of *truth*? After all,
> nominalists---people who don't believe in numbers, sets and so
on---all
> *accept* that AC is true iff every set has a choice set. I know a few
> prominent nominalists. They accept the truth condition of AC, but
deny that
> there are any sets. None of this has anything to do with truth.
(Nominalists
> tend to be disquotationalists about truth, but that's another
matter.)
>
> >You explicitly chosen "the real
> >world" to be the set of objects with respect to which the second
> >sentence is to be evaluated.
>
> Set of objects? I thought you didn't believe in sets!

I'm not committed to any position regarding the metaphysical status of
sets.  (As far as I know, anyway)


<snip>
> >
> >You're doing it again.  Where do these numbers live?
>
> How could a number "live somewhere".

This is a fairly standard informal usage asking what sorts of
structures support these objects.  Vectors live in vector spaces.
Numbers live in rings, not the same place as John Lennon does.

<snip>

>
> >Certainly not the
> >same place John Lennon does.
>
> OK. Progress. We have that:
> (a) John Lennon is (was) a physical object of some sort, and
> (b) Numbers are not physical objects.

> So, in the end, are you simply claiming that there aren't any
non-physical
> objects? In other words, are you trying to defend nominalism (or
formalism)?
> If so, then your ontology consists of physical objects, and this will
> include concrete expression tokens. And not, by the way, models,
which are
> abstract entities; also, not expression types, such as arbitrarily
long
> formulas; or proofs, which are arbitrarily long sequences of abstract
> expression types (formulas).

No, this isn't my position.  I'm not committed to any position
regarding the metaphysical status of the physical world or abstract
entities.

>
> Let's suppose that you want to be a nominalist. OK. If so, what does
this
> have to do with *truth*? Nothing, so far as I can tell. If there
aren't any
> non-physical objects, then the usual statements of mathematics
("there is an
> empty set", "there is an infinite set") are simply *false*. Indeed,
this is
> what my nominalist friends tell me.
>
> >By the way, the disquotational theory of
> >truth and the redundacy theory of truth are synonymous.
>
> They're not. The redundancy theory as usually understood was
advocated by
> Alfred Ayer (in 1936) and it is sometimes alleged that it was
advocated by
> Frank Ramsey in his "Facts and Propositions" paper, but this is
somewhat
> controversial, since Ramsey recognized a problem with generalizations
> expressed using the truth predicate.

OK.  You might want to suggest that Daniel Stoljar correct this in his
plato.stanford.edu entry on deflationary theories:

"The deflationary theory has gone by many different names, including at
least the following: the redundancy theory, the disappearance theory,
the no-truth theory, the disquotational theory, and the minimalist
theory. There is no terminological consensus about how to use these
labels: sometimes they are used interchangeably; sometimes they are
used to mark distinctions between different versions of the same
general view. "

<snip>

> >> (Proof: Let (L, I) be an interpreted language and let T be the set
of
> >truths
> >> in (L, I). Suppose that dom(I) contains all the expressions of L.
> >Suppose
> >> that L also contains a predicate True(x) which defines this set T.
> >For each
> >> expression E, let E* be a term in L such that E* denotes (in I) E.
> >Then, for
> >> any A in L, each sentence True(A*) <-> A is true in (L, I).
> >Disquotation is
> >> an intrinsic property of truth.
> >
> >A ha!   You're actually using the standard formulation of a model
> >theory to describe truth (via the disquotational-theory-esque
> >construction that a sentence S is true iff I(S) = T, where I is the
> >interpretation function).  This is how I would proceed.
>
> This may continue forever....
> There's an intuitive difference between:
>
> (a) truth for *interpreted* statements (Tarski 1933: "The Concept of
Truth
> in Formalized Languages")
> (b) truth-in-a-structure for *uninterpreted* formulas (Tarski 1935:
"On the
> Concept of Logical Consequence").

It's not particularly intuitive, since, in particular, once the
truth-in-a-structure of a sentence is determined, it is no longer
uninterpreted.  Furthermore, an interpreted statement is only in virtue
of its interpretation.  Under a different one, it's meaning (or in this
case, truth value) can change.  The two sets are co-extensive.

>
> It is the notion discussed in (a) that has the disquotational
property.
> When we speak, the statements we exchange are *interpreted*
statements ("Hi,
> I live in Scotand. My dog has no nose."), and the correct truth
predicate is
> (to a first approximation) disquotational. The second approximation
concerns
> vagueness, ambiguity, indexicality and intensional operators. This is
how
> real people actually use the truth predicate in their reasoning.
There's
> very little dispute about this.
>
> Now, there is an interpretation somehow attached or built-in to the
> interpreted statements we use. But no one knows how that works. It's
a major
> problem---maybe the central problem---in semantical theory. David
Lewis and
> Scott Soames (and Kripke in seminar talks) have suggested that an
> interpreted language is an abstract object. That's what I've called
(L, I).
> The problem is: what is it about a speaker's usage of L-expressions
that
> makes her a speaker of interpreted language (L, I) rather than (L,
I*)? What
> is it about my use of expressions that makes me a speaker of English
(where
> "cat" refers to cats) rather than Quinglish (where "cat" refers to
the
> mereological complements of cats)?
>
> Other people (of the more verificationist and Wittgensteinian
tradition)
> think that it involves an internal grasp of rules for reasoning with
and
> using (in particular, verifying and refuting) statements.

Count me among the Wittgensteinians.
>
> >But if you
> >don't fix I, you have no basis with which to interpret.  This is
> >EXACTLY my point.
>
> Right.
> But your central point is this: AC is an uninterpreted formula. And
you
> haven't given an argument for this sceptical claim (except maybe that
you
> don't know where sets "live").

Did you read the post that spawned this subthread?

> I don't know where sets "live" either, but
> that doesn't make me a radical meaning-sceptic.
> Curiously, however, you also think that meta-statements about models
are
> interpreted and assertible. Indeed, you assert them!
> It is these two assumptions which are incoherent (i.e., sentences
containing
> "set" are uninterpreted; sentences containing "model" are
interpreted).

Incoherent?  Says you, given your coarse understanding of what I've
written.  I suggest you read the Philosophical Investigations.
'cid 'ooh

0
poopdeville (133)
1/28/2005 5:43:31 AM
Jeffrey Ketland wrote:
> poopdeville@gmail.com wrote in message
> >Yes!  We need to know what symbols are, and what (the relevant)
> >syntactic rules are to do number theory.  But the syntactic rules
> >demonstrably don't pick out a unique interpretation, which is
exactly
> >why we need to be careful when talking about truth.
>
> Which "syntactic rules" do you have in mind?

The FOL, and things like addition, subtraction, multiplication,
exponentiation.

> Why do you believe that "syntactic rules" should "pick out a unique
> interpretation"?

This question is nonsensical.

> How exactly it is "demonstrated" that syntactic rules "don't pick out
a
> unique interpretation"?

By noting that there are different models of PA.  Godel's
incompleteness theorem is the easiest way to see that.

> What assumptions will you use in your
> "demonstration"?

1)  That people intend to use PA in a way that captures their
intuitions about numbers.
2)  The hypotheses of Godel's theorem.

>Are your assumptions true?

1)  Maybe.
2)  If you mean with respect to models of PA, yes.  They're true in
models of other paradigms of number theory, like intuitionist number
theory, too.

> What rules of reasoning will you
> use? Are they truth-preserving?

The rules of FO logic.  Yes.

> What is an interpretation? A set with a certain assignment function?

No, a mental picture of the objects one works with, as in Pinter.

> What is
> a set?

Don't know.  The most I'll commit to is "any object in a model of a set
theory."

> What is a function?

A set of ordered pairs of the form with the property that if the first
entries are equal, then so are the second.

> How do you define "A is true-in-interpretation I"?

With style.

> How do you define
> "formula"?

Look it up.  This is in Enderton's book.

> If "syntactic rules demonstrably don't pick out a unique
interpretation",
> then can you explain why this implies "we need to be careful when
talking
> about truth"? (It is consistent with the premise that the
interpretation is
> "picked out" some other way. How do you know?)

Because if we aren't careful, we'll have people with cavalier attitudes
and ideologies writing nonsense in books and articles.
>
> (It's easy to pretend to be a sceptic.)
But asking penetrating questions is hard.

'cid 'ooh

0
poopdeville (133)
1/28/2005 8:42:27 AM
poopdeville@gmail.com wrote in message

[lot of snippage]

>OK.  You might want to suggest that Daniel Stoljar correct this in his
>plato.stanford.edu entry on deflationary theories:
>
>"The deflationary theory has gone by many different names, including at
>least the following: the redundancy theory, the disappearance theory,
>the no-truth theory, the disquotational theory, and the minimalist
>theory. There is no terminological consensus about how to use these
>labels: sometimes they are used interchangeably; sometimes they are
>used to mark distinctions between different versions of the same
>general view. "

I need a lecture on deflationism? Cheers.
The above is OK (I read it many years ago, by the way). Deflationism is,
like I said, a broad church.

>It's not particularly intuitive, since, in particular, once the
>truth-in-a-structure of a sentence is determined, it is no longer
>uninterpreted.  Furthermore, an interpreted statement is only in virtue
>of its interpretation.  Under a different one, it's meaning (or in this
>case, truth value) can change.  The two sets are co-extensive.

Right. I agree with all of that. The truth value of "Lennon was left-handed"
depends upon "Lennon" referring to Lennon and "left-handed" referring to
left-handed things. As we know, the sentence is then false: because Lennon
was not left-handed. If "Lennon" had referred to Morrissey and "left-handed"
had referred to Mancunians, then it would have been true, since Morrissey is
a Mancunian.
There is an intimate connection between truth and meaning: roughly, if S
means that p, then S is true iff p.

You originally wondered what "AC is true" means, suggesting that it all
depended on the model. Tim Chow replied that it is equivalent to AC, and
further that it is equivalent to "AC is true in V". You seemed to be saying
that the equivalence between "AC is true" and AC was somehow mistaken. But
now you seem to concede all this. So, what was the point of the whole
detour? You now agree that AC is true iff every set of disjoint non-empty
sets has a choice set? Of course, if ZF is consistent, there is a model of
ZF in which AC is false. But this just says AC is not a theorem of ZF.
Unfortunately, this is not relevant to its truth value.
Similarly, a Goedel sentence G (for PA) is not a theorem of PA. But it is
true. Do we now require a huge detour concerning what "G is true" means? G
is true iff there is no number n such that n codes a proof in PA of G.

>Did you read the post that spawned this subthread?

If you mean a post by Tim Chow talking about "ZF is consistent", inter alia,
then yes. OK by me.

> I suggest you read the Philosophical Investigations.

I did, but l accidentally selected an "unintended interpretation" relative
to which all of Ludwig's statements turned out false!

Counter-suggestion: if you have strong views (as you seem to) about the
notion of truth, it might help to read some of the modern literature on
truth, including the anthology by Michael Lynch (2001), the Hartry Field
volume (_Truth and the Absence of Fact_, 2001) and the volume edited by
Halbach and Horsten (2002), which contains a lot of useful technical work.

--- Jeff


0
ketland (18)
1/28/2005 8:44:27 AM
PS - nevermind the sentence that reads "In my understanding..."  I try
to be careful about writing these things, so I do multiple drafts on
occassion.  A remnant of a previous draft remained.  :-)

'cid 'ooh

0
poopdeville (133)
1/28/2005 8:44:28 AM
poopdeville@gmail.com wrote in message

>Because if we aren't careful, we'll have people with cavalier attitudes
>and ideologies writing nonsense in books and articles.

Cavalier atttitudes and ideologies! Writing nonsense! In books and articles!
This is outrageous! Off to the dungeons with them all!

--- Jeff


0
ketland (18)
1/28/2005 9:10:23 AM
poopdeville@gmail.com writes:

> > Why do you believe that "syntactic rules" should "pick out a unique
> > interpretation"?
> 
> This question is nonsensical.

  Why is the question nonsensical? You said that the syntactic rules
do not pick out a unique interpretation, so it seems that you should mean
something by "the syntactic rules pick out a unique interpretation".
0
torkel (478)
1/28/2005 3:44:46 PM
Torkel Franzen wrote:
> poopdeville@gmail.com writes:
>
> > > Why do you believe that "syntactic rules" should "pick out a
unique
> > > interpretation"?
> >
> > This question is nonsensical.
>
>   Why is the question nonsensical? You said that the syntactic rules
> do not pick out a unique interpretation, so it seems that you should
mean
> something by "the syntactic rules pick out a unique interpretation".

Because it presupposes that I want syntactic rules to pick out unique
interpretations.  

'cid 'ooh

0
poopdeville (133)
1/28/2005 4:32:14 PM
In article <1106877661.601587.62140@f14g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Now that we have a common language to work with, many of my claims are
>uncontroversial.  However, we've both brought some philosophical
>baggage to the table.  In my understanding of your position, you're
>committed to the existence of sets.

To make the main point that truth-in-a-model isn't, and can't be, the
only notion of truth, I don't need to be committed to the existence
of sets.

For me to make statements such as "the cartesian product of nonempty
sets is nonempty" and claim that I'm saying something meaningful, I
do indeed need to be committed to the existence of sets in some sense.
But I don't have to be committed to any kind of platonism.  More to
the point, I'm not really committed to the existence of sets any more
than *you* are when you say, "ZFC is consistent iff ZFC has models"
(and think that you're saying something meaningful), because a model is
a set.

>Yes!  We need to know what symbols are, and what (the relevant)
>syntactic rules are to do number theory.  But the syntactic rules
>demonstrably don't pick out a unique interpretation, which is exactly
>why we need to be careful when talking about truth.

To respond to this let me skip forward a bit to:

>If you can tell me what a set is, outside of a "collection" (the
>more-or-less personal "mental picture") or "an element of a collection
>which satisfies the membership relation" (the public notion that
>doesn't pick out one of these mental pictures), that'd be great.  :-)

I'll tell you what a set is if you can tell me what a symbol is.

By the way, historically, symbols were often thought of as being less well
understood than integers.  The whole notion of Goedel numbering is, in a
sense, a symptom of the feeling that we know how to deal with integers, so
to convince ourselves that we know how to deal with symbols and strings,
let's convert symbols and strings to integers, so that we *really* know
what we're talking about.

But that's an aside.  Let me take your argument about unique interpretations
and turn it against you.  Logic, like any other branch of mathematics, can
be formalized and studied.  In particular, we can develop a first-order
theory of syntax, which captures discourse about symbols and strings and
so forth.  When we do this, we discover that sentences about symbols,
strings, rules, and so forth admit nonstandard interpretations.  We might
*think* we know what we mean by a proof of finite length, but lo and behold,
nonstandard models of axioms for syntax show us that "finite length" proofs
could be "nonstandardly finite," which is what most people would call
infinite.  Does this shake your belief that you know what symbols and proofs
are?

>> The set of axioms for a group does not mimic the *discourse* of group
>> theorists.
[...]
>I had not thought of this distinction.  

Let me elaborate a bit because I think this exercise can be helpful.
Suppose we try to express "every finite simple group is X" in the
first-order language of groups.  (I'm being lazy and not saying what
X is, but as you'll see, we won't need to know anything about X other
than that it's some property that groups might have.)  As a first step
towards formalization we might try to rephrase:

  For all groups G, if G is finite and G is simple, then G is X.

The problem is that we're quantifying over *groups* here ("for all
groups G"), and in the first-order language of groups we can only
quantify over *group elements*.  (That's what "first-order" *means*.)

Let's not give up just yet, though.  If we could find a sentence
"FiniteSimple" in the first-order language of groups such that
"GroupAxioms & FiniteSimple" is satisfied by all and only the finite
simple groups, then

   GroupAxioms & FiniteSimple  ->  X

would be a tolerably good formalization.  The problem, though, is
the basic theorem of mathematical logic that any first-order sentence
that is satisfiable by arbitrarily large finite structures is also
satisfiable by some infinite structures.  So we can't write down a
sentence that captures "finite" in the sense we want.

Problems also arise with "simple" because this means that there are no
nontrivial normal subgroups.  A subgroup is a *subset* of the group,
and again we can't talk directly about subsets in a first-order
language, only about group elements.

These limitations illustrate that when people study the first-order
language of groups, they're *not* trying to mimic a large fraction of
group-theoretic discourse; they're interested specifically in *first-order
properties of groups*, which is a very limited fraction of the properties
of groups that group theorists in general are interested in.

In contrast, when people study ZFC, they are often investigating the
foundations of mathematics as a whole, and it's important that most of
mathematical discourse can be mimicked more-or-less directly in the
language.  It's not that we're interested in---for lack of a better
term---"universes" and we're trying to isolate the first-order properties
of universes for special scrutiny.  We're actually trying to find formal
counterparts for all the kinds of statements that mathematicians make.

The two projects are quite different and the fact that the same tool
(first-order logic) is being used in both cases does not mean that
direct parallels between ZFC and the axioms for a group always make
sense.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/28/2005 4:37:47 PM
poopdeville@gmail.com writes:

> Because it presupposes that I want syntactic rules to pick out unique
> interpretations.  

  So what does it mean for syntactic rules to pick out a unique
interpretation? Is there such a thing?

0
torkel (478)
1/28/2005 4:44:25 PM
poopdeville@gmail.com wrote in message
>> If "syntactic rules demonstrably don't pick out a unique
>interpretation",
>> then can you explain why this implies "we need to be careful when
>talking
>> about truth"? (It is consistent with the premise that the
>interpretation is
>> "picked out" some other way. How do you know?)
>
>Because if we aren't careful, we'll have people with cavalier attitudes
>and ideologies writing nonsense in books and articles.

I'm still finding your philosophical view quite puzzling.
It seems to have the following overall reductionist structure:

(i) there are notions D1, D2, D3, ..., that you consider determinate.
(ii) there are notions M1, M2, M3, ... that you find meaningless,
indeterminate or model-relative (unless they can be reduced to kosher
notions in (i)).

Then you classify these as follows

    Determinate notions              Meaningless, indeterminate notions
    "syntactic rule",                         "set"
    "formula",                                   "truth simpliciter"
     "model",                                    "natural number"
    "true-in-an-interpretation".

Your arguments (correct me if I'm wrong!) for this classification go roughly
as follows:

1. The notion of "set" is indeterminate (or model-relative). This is because
the "syntactic rules" (a notion you accept as kosher) for ZF don't "pick out
a unique interpretation".
2. The notion of "truth simpliciter" is meaningless. This is because the
*only way* that truth simpliciter could be a kosher notion would be via
reductive analysis: i.e., if the "syntactic rules" of the language "picked
out a unique interpretation", then truth simpliciter would be truth in this
unique model.
3. The notion of "natural number" is indeterminate (or model-relative). This
is because there are non-standard models of the "syntactic rules" of Peano
arithmetic.

Is that it?
(And, though it's not too important, you think that people who don't agree
with your philosophical views are not "careful" and have "cavalier attitudes
and ideologies" and they "write nonsense". Kurt Goedel, for example, "wrote
nonsense". Similarly, Hugh Woodin and others.)

--- Jeff


0
ketland (18)
1/28/2005 8:54:34 PM
Jeffrey Ketland wrote:
> poopdeville@gmail.com wrote in message
> >> If "syntactic rules demonstrably don't pick out a unique
> >interpretation",
> >> then can you explain why this implies "we need to be careful when
> >talking
> >> about truth"? (It is consistent with the premise that the
> >interpretation is
> >> "picked out" some other way. How do you know?)
> >
> >Because if we aren't careful, we'll have people with cavalier
attitudes
> >and ideologies writing nonsense in books and articles.
>
> I'm still finding your philosophical view quite puzzling.
> It seems to have the following overall reductionist structure:
>
> (i) there are notions D1, D2, D3, ..., that you consider determinate.
> (ii) there are notions M1, M2, M3, ... that you find meaningless,
> indeterminate or model-relative (unless they can be reduced to kosher
> notions in (i)).
>
> Then you classify these as follows
>
>     Determinate notions              Meaningless, indeterminate
notions
>     "syntactic rule",                         "set"
>     "formula",                                   "truth simpliciter"
>      "model",                                    "natural number"
>     "true-in-an-interpretation".
>
> Your arguments (correct me if I'm wrong!) for this classification go
roughly
> as follows:
>
> 1. The notion of "set" is indeterminate (or model-relative). This is
because
> the "syntactic rules" (a notion you accept as kosher) for ZF don't
"pick out
> a unique interpretation".

Correct.

> 2. The notion of "truth simpliciter" is meaningless. This is because
the
> *only way* that truth simpliciter could be a kosher notion would be
via
> reductive analysis: i.e., if the "syntactic rules" of the language
"picked
> out a unique interpretation", then truth simpliciter would be truth
in this
> unique model.

More-or-less correct, though I'll comment on this point in a bit.

> 3. The notion of "natural number" is indeterminate (or
model-relative). This
> is because there are non-standard models of the "syntactic rules" of
Peano
> arithmetic.
>

Yes, though this is just an instance of a more general phenomenon, as
is 1.

> Is that it?

Given the context, yes -- with the caveat that formulae are
undeterminate in content until interpreted.  But as raw strings of
symbols, they belong on the left.

We can call the left column above "rule-or-symbol-like" and the right
"object-like."  The elements in the left column include precisely what
is necessary to communicate:  symbols (in the form of formulae) and
rules for interpretation (in the form of models and their associated
machinery).

Someone working with sets or natural numbers would have some intuition
about their relations, but in the absence of an axiomatization of the
relations that picks out a unique model (in the logical sense), our
mathematician would not be able to speak (that is, use the language of
his axiomatization) of his intuitive concepts exclusively.  Following
Wittgenstein, of what one cannot speak, one should remain silent.
'cid 'ooh

0
poopdeville (133)
1/29/2005 1:38:25 AM
poopdeville@gmail.com wrote in message
>>     Determinate notions              Meaningless, indeterminate
>notions
>>     "syntactic rule",                         "set"
>>     "formula",                                   "truth simpliciter"
>>      "model",                                    "natural number"
>>     "true-in-an-interpretation".
>>
>> Your arguments (correct me if I'm wrong!) for this classification go
>roughly
>> as follows:
>>
>> 1. The notion of "set" is indeterminate (or model-relative). This is
>because
>> the "syntactic rules" (a notion you accept as kosher) for ZF don't
>"pick out
>> a unique interpretation".
>
>Correct.
>
>> 2. The notion of "truth simpliciter" is meaningless. This is because
>the
>> *only way* that truth simpliciter could be a kosher notion would be
>via
>> reductive analysis: i.e., if the "syntactic rules" of the language
>"picked
>> out a unique interpretation", then truth simpliciter would be truth
>in this
>> unique model.
>
>More-or-less correct, though I'll comment on this point in a bit.
>
>> 3. The notion of "natural number" is indeterminate (or
>model-relative). This
>> is because there are non-standard models of the "syntactic rules" of
>Peano
>> arithmetic.
>>
>
>Yes, though this is just an instance of a more general phenomenon, as
>is 1.
>
>> Is that it?
>
>Given the context, yes -- with the caveat that formulae are
>undeterminate in content until interpreted.  But as raw strings of
>symbols, they belong on the left.
>
>We can call the left column above "rule-or-symbol-like" and the right
>"object-like."  The elements in the left column include precisely what
>is necessary to communicate:  symbols (in the form of formulae) and
>rules for interpretation (in the form of models and their associated
>machinery).
>
>Someone working with sets or natural numbers would have some intuition
>about their relations, but in the absence of an axiomatization of the
>relations that picks out a unique model (in the logical sense), our
>mathematician would not be able to speak (that is, use the language of
>his axiomatization) of his intuitive concepts exclusively.  Following
>Wittgenstein, of what one cannot speak, one should remain silent.
>'cid 'ooh

OK. Let's get back to your distinction between the determinate notions and
the indeterminate ones, which I organized as follows:

Determinate notions              Meaningless/indeterminate notions
    "syntactic rule",                         "set"
    "formula",                                   "truth simpliciter"
     "model",                                    "natural number"
   "true-in-an-interpretation".

The two arguments for indeterminacy go:

1. The notion of "set" (or "number") is indeterminate (or model-relative).
This is because there are non-standard models. That is, the (first-order)
"syntactic rules" (a notion you accept as kosher) for ZF (or PA) don't pick
out a unique interpretation.

2. The notion of "truth simpliciter" is meaningless. This is because the
*only way* that truth simpliciter could be a kosher notion would be via
reductive analysis: i.e., if the "syntactic rules" of the language "picked
out a unique interpretation", then truth simpliciter would be truth in this
unique model.

(A sub-objection to (2) was that the disquotational T-scheme as analysed by
Tarski 1933 provides a perfectly serviceable and correct truth predicate
which is not relativized. Incidentally, see the Investigations, para 136, by
the way, for Ludwig's view. But never mind that.)

Now the main objection to all this just uses a simple tu quoque, which Tim
Chow has also explained in parallel posts. Your own arguments imply that
you're contradicting yourself in thinking that "formula", "model", etc., are
determinate.

Here is a version of your definition of "determinate notion":

  K is a determinate notion iff there is a system of (first-order) syntactic
  rules and axioms for K which is categorical.

Using the very same methods which you yourself consider acceptable for the
other cases ("set" and "number"), the following are mathematical theorems
which also follow from your definition:

(i) The notion "syntactic rule" is indeterminate.
(ii) The notion "formula" is indeterminate.
(iii) the notion "model" is indeterminate.
(iv) the notion "true-in-an-interpretation" is indeterminate.

(Actually, the notion "finite" is indeterminate as well, by a similar
argument.)

So, the classification above is wrong. *All* the notions ("formula",
"syntactic rule", "model", etc.) which you believe to be determinate, can in
fact be demonstrated to be *indeterminate*---on your proposed definition of
determinate---using exactly the same methods that you insist show that
"set", "truth simpliciter" and "number" are indeterminate. Consistency
requires that you should junk all of them.

>Following Wittgenstein, of what one cannot speak, one should remain silent.

OK. Junk them all then. I assume that you will now "remain silent". For it
follows from this prescription (plus the above facts about indeterminacy)
that, since you are consistent, you should now "remain silent". If you do
not remain sillent, I will conclude that you reject Wittgenstein's
prescription. Then people will say: why you think that Wittgenstein was
wrong?

--- Jeff


0
ketland (18)
1/29/2005 2:41:58 AM
Torkel Franzen <torkel@sm.luth.se> writes:

>poopdeville@gmail.com writes:
>
>> Because it presupposes that I want syntactic rules to pick out unique
>> interpretations.  
>
>  So what does it mean for syntactic rules to pick out a unique
>interpretation? Is there such a thing?

Is the answer to "Is there such a thing as a unique interpretation?" 
perhaps to be found in the commentary on St. Anselm's logical work?

Lee Rudolph

0
lrudolph (277)
1/29/2005 11:16:55 AM
Torkel Franzen wrote:
> poopdeville@gmail.com writes:
>
> > Because it presupposes that I want syntactic rules to pick out
unique
> > interpretations.
>
>   So what does it mean for syntactic rules to pick out a unique
> interpretation? Is there such a thing?

A set of syntactic rules is more-or-less a set of transformations
converting one un-interpreted symbol into another.  In this context we
might consider a particular formalization of the FOL and a set of
axioms for a language as a set of syntactic rules.  A set of syntactic
rules, then, picks out a unique interpretation if there is a unique
model of the syntactic rules as explained here.

A simple example requires that we consider syntactic rules over SOL,
but this is inessential to my point:  all complete ordered fields are
isomorphic.

'cid 'ooh

0
poopdeville (133)
1/29/2005 11:51:49 AM
poopdeville@gmail.com writes:

> A set of syntactic rules is more-or-less a set of transformations
> converting one un-interpreted symbol into another.  In this context we
> might consider a particular formalization of the FOL and a set of
> axioms for a language as a set of syntactic rules.  A set of syntactic
> rules, then, picks out a unique interpretation if there is a unique
> model of the syntactic rules as explained here.

> A simple example requires that we consider syntactic rules over SOL,
> but this is inessential to my point:  all complete ordered fields are
> isomorphic.

  They are indeed, but what does this have to do with syntactic rules?
That all complete ordered fields are isomorphic is a set-theoretical
theorem. It is not a statement that there is a unique model of any
"syntactic rules". Indeed there is no standard notion in logic of a
model of a set of syntactic rules in your sense of a set of
"transformations converting one uninterpreted symbol into
another". That all complete ordered fields are isomorphic means that
all models of a certain second order theory, that is, all structures
in which the axioms of the theory are *true*, are isomorphic. How
would you consider the axioms of the theory as "transformations
converting..." etc?
0
torkel (478)
1/29/2005 12:09:50 PM
tchow@lsa.umich.edu says...

>To make the main point that truth-in-a-model isn't, and can't be, the
>only notion of truth, I don't need to be committed to the existence
>of sets.
>
>For me to make statements such as "the cartesian product of nonempty
>sets is nonempty" and claim that I'm saying something meaningful, I
>do indeed need to be committed to the existence of sets in some sense.
>But I don't have to be committed to any kind of platonism.

I don't understand why not. What is there to mathematical
platonism other than commitment to the existence of sets?

>More to the point, I'm not really committed to the existence of
>sets any more than *you* are when you say, "ZFC is consistent
>iff ZFC has models" (and think that you're saying something
>meaningful), because a model is a set.

I don't understand why talk of models *isn't* a commitment to
the existence of sets.

I think it's a little hazy in what sense sets exist or don't
exist. We can certainly talk about them meaningfully *as* if
they exist, but that doesn't require anything any more than
some kind of coherence of our story about them.

--
Daryl McCullough
Ithaca, NY

0
1/29/2005 1:46:29 PM
In article <ctg43l01cjo@drn.newsguy.com>,
Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
>I don't understand why not. What is there to mathematical
>platonism other than commitment to the existence of sets?

First of all, I'll say that personally I lean towards platonism.  I didn't
always used to be this way; many years ago I was much more influenced by
formalism, but have gradually changed my point of view.  So *my* commitment
to the existence of sets is tinged with platonism.

That aside, I would answer your question by quoting something you said later
in the same article:

>I think it's a little hazy in what sense sets exist or don't
>exist. We can certainly talk about them meaningfully *as* if
>they exist, but that doesn't require anything any more than
>some kind of coherence of our story about them.

As you say, committing to the existence of sets is something of a hazy
commitment until the nature of this "existence" is fleshed out a bit.
I can commit to the existence of sets, yet flesh out the details in
different ways.  I might adopt a fictionalist posture, somewhat like
what you describe here.  That is, I regard my statements as meaningful
*according to a certain story*.  Oliver Twist exists according to a
certain story; this is a meaningful statement and not just a syntactic
entity to be manipulated according to syntactic rules, yet Oliver Twist
is of course a fictional character.

A platonist doesn't regard sets as "useful fictions" but as real, and
hence is likely to say that meaningful statements about sets are either
true or false, whereas a fictionalist is usually happy to say that some
statements are intelligible ("Oliver Twist's left thumbprint was a whorl")
but have indeterminate truth value.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/29/2005 6:23:21 PM
In article <41fbd499$0$557$b45e6eb0@senator-bedfellow.mit.edu>,
	tchow@lsa.umich.edu writes:
>In article <ctg43l01cjo@drn.newsguy.com>,
>Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
>>I don't understand why not. What is there to mathematical
>>platonism other than commitment to the existence of sets?
>
>First of all, I'll say that personally I lean towards platonism.  I didn't
>always used to be this way; many years ago I was much more influenced by
>formalism, but have gradually changed my point of view.  So *my* commitment
>to the existence of sets is tinged with platonism.
>
>That aside, I would answer your question by quoting something you said later
>in the same article:
>
>>I think it's a little hazy in what sense sets exist or don't
>>exist. We can certainly talk about them meaningfully *as* if
>>they exist, but that doesn't require anything any more than
>>some kind of coherence of our story about them.
>
>As you say, committing to the existence of sets is something of a hazy
>commitment until the nature of this "existence" is fleshed out a bit.
>I can commit to the existence of sets, yet flesh out the details in
>different ways.  I might adopt a fictionalist posture, somewhat like
>what you describe here.  That is, I regard my statements as meaningful
>*according to a certain story*.  Oliver Twist exists according to a
>certain story; this is a meaningful statement and not just a syntactic
>entity to be manipulated according to syntactic rules, yet Oliver Twist
>is of course a fictional character.
>
>A platonist doesn't regard sets as "useful fictions" but as real, and
>hence is likely to say that meaningful statements about sets are either
>true or false, whereas a fictionalist is usually happy to say that some
>statements are intelligible ("Oliver Twist's left thumbprint was a whorl")
>but have indeterminate truth value.

Hmmm! Well I think of myself as a platonist, but I don't consider that
it is meaningful to say that the axiom of choice, which is a meaningful
statement about sets, is either true or false in any absolute sense.
So perhaps I am being inconsistent? I find it strange that there are
some mathematicians who do claim to believe that ACC is true or false,
although they do not generally expect ever to find out which!

Derek Holt.
0
mareg (22)
1/29/2005 6:38:03 PM
In article <ctgl6b$bck$1@wisteria.csv.warwick.ac.uk>,
 <mareg@mimosa.csv.warwick.ac.uk> wrote:
>Hmmm! Well I think of myself as a platonist, but I don't consider that
>it is meaningful to say that the axiom of choice, which is a meaningful
>statement about sets, is either true or false in any absolute sense.
>So perhaps I am being inconsistent?

I did say that a platonist was "likely" to believe in bivalence.  I
don't think it's necessarily inconsistent to reject bivalence and
embrace platonism.  Platonism is what I would call "mathematical realism"
and there many variations of realism.

>I find it strange that there are
>some mathematicians who do claim to believe that ACC is true or false,
>although they do not generally expect ever to find out which!

Well, "there is an earthlike planet in a distant galaxy with intelligent
life" is surely either true or false (after some of the vaguer words in
that sentence are made more precise), but many people do not expect to
find out which.

I think what makes the mathematical situation seem strange to many people
is that they find it hard to envision under what circumstances, even ideal
circumstances, they would come to believe that (say) the continuum
hypothesis is true.  At least in the case of the distant planet, or even
the consistency of various axiomatic systems, they can envisage an idealized
situation under which they would have enough evidence to embrace the claim.
But somehow the "ideal situation" in which one has transfinite powers that
allow searching through *all* bijections for a counterexample to the
continuum hypothesis seems *too* idealized for comfort.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/29/2005 7:04:33 PM
<mareg@mimosa.csv.warwick.ac.uk> ha scritto

> Hmmm! Well I think of myself as a platonist, but I don't consider that
> it is meaningful to say that the axiom of choice, which is a meaningful
> statement about sets, is either true or false in any absolute sense.
> So perhaps I am being inconsistent? I find it strange that there are
> some mathematicians who do claim to believe that ACC is true or false,
> although they do not generally expect ever to find out which!

Is there any statement about set that you consider true or false in some
absolute sense?



0
1/29/2005 7:14:27 PM
mareg@mimosa.csv.warwick.ac.uk wrote:

> Hmmm! Well I think of myself as a platonist, but I don't consider that
> it is meaningful to say that the axiom of choice, which is a meaningful
> statement about sets, is either true or false in any absolute sense.
> So perhaps I am being inconsistent? I find it strange that there are
> some mathematicians who do claim to believe that ACC is true or false,
> although they do not generally expect ever to find out which!

I'm not sure whom you would put in the category described.  Most
mathematicians who are realists about sets are convinced that AC
is true.

(I'm assuming that "ACC" was a typo for "AC"; if not, then I
suppose my response could be a non-sequitur.)

0
mike_lists (73)
1/29/2005 7:15:21 PM
In article <3625maF4pqgpgU1@individual.net>,
	Mike Oliver <mike_lists@verizon.net> writes:
>mareg@mimosa.csv.warwick.ac.uk wrote:
>
>> Hmmm! Well I think of myself as a platonist, but I don't consider that
>> it is meaningful to say that the axiom of choice, which is a meaningful
>> statement about sets, is either true or false in any absolute sense.
>> So perhaps I am being inconsistent? I find it strange that there are
>> some mathematicians who do claim to believe that ACC is true or false,
>> although they do not generally expect ever to find out which!
>
>I'm not sure whom you would put in the category described.  Most
>mathematicians who are realists about sets are convinced that AC
>is true.
>
>(I'm assuming that "ACC" was a typo for "AC"; if not, then I
>suppose my response could be a non-sequitur.)

Yes, I meant "AC". My impression is rather that most mathematicians prefer
to assume AC (rather than not AC), because it results in cleaner statements
of theorems, like every vector space has a basis. I generally assume and use
AC myself, but that is not the same thing as believing that it is any more
true than not AC. What I find surprising is that there are people who
believe that it is either true or false, but they don't know which.

Derek Holt.


0
mareg (22)
1/29/2005 8:34:11 PM
tchow@lsa.umich.edu says...

>As you say, committing to the existence of sets is something of a hazy
>commitment until the nature of this "existence" is fleshed out a bit.
>I can commit to the existence of sets, yet flesh out the details in
>different ways.  I might adopt a fictionalist posture, somewhat like
>what you describe here.  That is, I regard my statements as meaningful
>*according to a certain story*.  Oliver Twist exists according to a
>certain story; this is a meaningful statement and not just a syntactic
>entity to be manipulated according to syntactic rules, yet Oliver Twist
>is of course a fictional character.
>
>A platonist doesn't regard sets as "useful fictions" but as real

That's sort of what is at issue: what does it *mean* to regard them
as real? How would anything be any different if they were real or not?

Is there really any commitment involved, or is it just a difference
in the way of speaking?

>and hence is likely to say that meaningful statements about sets are either
>true or false, whereas a fictionalist is usually happy to say that some
>statements are intelligible ("Oliver Twist's left thumbprint was a whorl")
>but have indeterminate truth value.

But isn't this just a difference in the feeling of "completeness" of
the stories? We feel that our description of sets (the cumulative
hierarchy) are categorical, while Dickens characterization of Oliver
Twist is not. But does there have to be a real object in mind in order
for a description to be categorical?

--
Daryl McCullough
Ithaca, NY

0
1/29/2005 8:58:08 PM
mareg@mimosa.csv.warwick.ac.uk wrote:

> My impression is rather that most mathematicians prefer
> to assume AC (rather than not AC), because it results in cleaner statements
> of theorems, like every vector space has a basis. I generally assume and use
> AC myself, but that is not the same thing as believing that it is any more
> true than not AC. What I find surprising is that there are people who
> believe that it is either true or false, but they don't know which.

So do you have any names of people who think that?

I continue to think that far more common than the position you
mention in your last sentence -- and indeed the overwhelming
majority opinion among realists about sets -- is that AC is
really true.

Substitute "CH" for "AC" and yes, there are probably a
great many who think it's really true or really false, but
who have no strong expectation of ever finding out which.

0
mike_lists (73)
1/29/2005 10:15:51 PM
In article <ctgtd00tjt@drn.newsguy.com>,
Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
>That's sort of what is at issue: what does it *mean* to regard them
>as real? How would anything be any different if they were real or not?
>
>Is there really any commitment involved, or is it just a difference
>in the way of speaking?

I suppose that depending on how good an "actor" you are, it might be
difficult to distinguish between a fictionalist and a realist.  However,
let's take Hugh Woodin as an example.  Woodin certainly believes that CH is
either true or false, and even holds out hope of figuring out which it
is.  As far as I can tell, this stems from his realism about sets.  This
drives his research program.

You could argue that someone else who is not a realist about sets could
publish exactly the same theorems as Woodin, and even say the same things
about the truth or falsity of CH (and projective determinacy), all the while
privately adding "according to a certain story" to everything.  I would find
that rather odd, though.  I can't imagine Ed Nelson, who spends some of his
time searching for contradictions in PA, pursuing Woodin's research program.
(Nelson is a formalist rather than a fictionalist, but you see my point.)
So I think there is a commitment involved.

>But isn't this just a difference in the feeling of "completeness" of
>the stories? We feel that our description of sets (the cumulative
>hierarchy) are categorical, while Dickens characterization of Oliver
>Twist is not.

That could be.  But off the top of my head, it seems that everything we
uncontroversially regard as a "fictional creation" fails to be "complete"
for the simple reason that a finite human creator can't possibly flesh
out every last detail.  Bivalence seems to me to be very naturally tied
to the notion of a real existence independent from us, even if strictly
speaking it doesn't have to be.
-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences
0
tchow (869)
1/30/2005 1:13:27 AM
In article <41fc34b7$0$572$b45e6eb0@senator-bedfellow.mit.edu>,
 tchow@lsa.umich.edu wrote:

>In article <ctgtd00tjt@drn.newsguy.com>,
>Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
[snip]

>>But isn't this just a difference in the feeling of "completeness" of
>>the stories? We feel that our description of sets (the cumulative
>>hierarchy) are categorical, while Dickens characterization of Oliver
>>Twist is not.
>
>That could be.  But off the top of my head, it seems that everything we
>uncontroversially regard as a "fictional creation" fails to be "complete"
>for the simple reason that a finite human creator can't possibly flesh
>out every last detail.  Bivalence seems to me to be very naturally tied
>to the notion of a real existence independent from us, even if strictly
>speaking it doesn't have to be.

Well, 2nd-order PA is categorical for the natural numbers, but many would 
argue against their independent existence.  ISTM a Platonist (or other 
Mathematical Realist) has something more significant in mind than just 
categoricity.

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------
0
see80 (286)
1/30/2005 1:58:31 AM
Tim Chow writes:
|You could argue that someone else who is not a realist about sets
could
|publish exactly the same theorems as Woodin, and even say the same
things
|about the truth or falsity of CH (and projective determinacy), all the
while
|privately adding "according to a certain story" to everything.  I
would find
|that rather odd, though.  I can't imagine Ed Nelson, who spends some
of his
|time searching for contradictions in PA, pursuing Woodin's research
program.
|(Nelson is a formalist rather than a fictionalist, but you see my
point.)
|So I think there is a commitment involved.

I wish I remembered for sure which formalist it was, but I seem
to remember reading some set theory done by a formalist, where
he described the most popular position as being that there exist
ordinals alpha such that V_alpha satisfies the same first-order
sentences as V (and I think the typical philosophical-realist set
theorist does think this), but that he preferred some axioms that
contradicted that. It was sort of unusual.

I remember seeing a paper of Nelson's on nonstandard analysis
where he developed it in a conservative extension of something
like ZFC, extended with a predicate "is standard" for which there
are some additional axioms.

I think it can be very hard to say just what kind of commitment a
realist or Platonist is making. The fact that one doesn't feel the
need to relativize things to models or theories is maybe the main
practical effect.

Keith Ramsay

0
kramsay (59)
1/30/2005 3:31:21 AM
"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> ha scritto

> > Hmmm! Well I think of myself as a platonist, but I don't consider that
> > it is meaningful to say that the axiom of choice, which is a meaningful
> > statement about sets, is either true or false in any absolute sense.
> > So perhaps I am being inconsistent? I find it strange that there are
> > some mathematicians who do claim to believe that ACC is true or false,
> > although they do not generally expect ever to find out which!
>
> Is there any statement about set that you consider true or false in some
> absolute sense?

What I wanted to point out is this:

There are 2 possibilities:

1) you think that statements about sets in general cannot be considered true
or false (in any absolute sense).
In this case I think it is not clear in which sense you could be "platonist"
because you would find that "there exists the empty set" or "there exist the
set of natural number" are not true (nor false) statements (in any absolutse
sense), while platonism is (usually) characterized by belief in existence of
mathematical objects.

2) you think that some statements about sets are true or false and some are
not (i.a.a.s.), so it is not clear why AC is a special case, what makes the
case of AC different from (for example) the axiom of infinity so thet the
former has not an (absolute) truth value while the latter does have?


0
1/30/2005 8:43:51 AM
tchow@lsa.umich.edu says...

>Daryl McCullough <stevendaryl3016@yahoo.com> wrote:

>>But isn't this just a difference in the feeling of "completeness" of
>>the stories? We feel that our description of sets (the cumulative
>>hierarchy) are categorical, while Dickens characterization of Oliver
>>Twist is not.
>
>That could be.  But off the top of my head, it seems that everything we
>uncontroversially regard as a "fictional creation" fails to be "complete"
>for the simple reason that a finite human creator can't possibly flesh
>out every last detail.  Bivalence seems to me to be very naturally tied
>to the notion of a real existence independent from us, even if strictly
>speaking it doesn't have to be.

So this kind of platonism says that anything that can be described in
enough detail as to answer (in principle, if not in practice) all
possible questions has a real existence. I guess that makes sense,
although it seems to me that belief in the categoricity of a description
is all you need for most "applications" of platonism (such as whether
it is meaningful to say that CH is true or false).

--
Daryl McCullough
Ithaca, NY

0
1/30/2005 2:03:40 PM
Daryl McCullough wrote:
> tchow@lsa.umich.edu says...
>
> >Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
>
> >>But isn't this just a difference in the feeling of "completeness"
of
> >>the stories? We feel that our description of sets (the cumulative
> >>hierarchy) are categorical, while Dickens characterization of
Oliver
> >>Twist is not.
> >
> >That could be.  But off the top of my head, it seems that everything
we
> >uncontroversially regard as a "fictional creation" fails to be
"complete"
> >for the simple reason that a finite human creator can't possibly
flesh
> >out every last detail.  Bivalence seems to me to be very naturally
tied
> >to the notion of a real existence independent from us, even if
strictly
> >speaking it doesn't have to be.
>
> So this kind of platonism says that anything that can be described in
> enough detail as to answer (in principle, if not in practice) all
> possible questions has a real existence. I guess that makes sense,
> although it seems to me that belief in the categoricity of a
description
> is all you need for most "applications" of platonism (such as whether
> it is meaningful to say that CH is true or false).
>

Interesting view on CH, especially considering the fact that there can
be no such thing as continuum in the physical world.

--
Eray

0
examachine (384)
2/2/2005 2:05:28 AM
examachine@gmail.com wrote:
> 
> Interesting view on CH, especially considering the fact that there can
> be no such thing as continuum in the physical world.

That's a non sequitur in at least 2 ways.

1) CH is about cardinals and ordinals, not about the continuum.

2) how can you say "there -can- be no continuum" in the physical 
world? The continuum is a concept. There also can't be a 2 in the 
physical world.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
2/2/2005 9:22:43 AM
Mitch Harris wrote:
> examachine@gmail.com wrote:
> >
> > Interesting view on CH, especially considering the fact that there
can
> > be no such thing as continuum in the physical world.
>
> That's a non sequitur in at least 2 ways.
>
> 1) CH is about cardinals and ordinals, not about the continuum.

Really. What does the continuum hypothesis say about the continuum?

> 2) how can you say "there -can- be no continuum" in the physical
> world? The continuum is a concept. There also can't be a 2 in the
> physical world

2 is a number. We observe many *copies* of this concept in the physical
world, e.g. there are such things as binary quantum states. We observe
no such thing as actual infinite divisibility in the physical world,
there is no such thing as a continuous space as the property of _any_
subset of space-time. It simply does not exist, and furthermore in a
mechanical world like ours it _cannot_ exist. This is a deep
metaphysical difference between an integer and a _real_ number.

One would have to forget the obsolete platonist way of talking about
mathematics if one would like to understand the difference. There can
be no such thing as "the truth" to the CH, I basically think
Kronecker's comments on this problem were exactly true. This shows
itself, but those who have some kind of sympathy to naive set theory,
naive geometry and naive number theory want to hold on to these
notions. It seems like an emotional reaction of some sort.
Regards,

--
Eray Ozkural

0
examachine (384)
2/2/2005 12:21:43 PM
examachine@gmail.com wrote:
> Mitch Harris wrote:
>>examachine@gmail.com wrote:
>>
>>>Interesting view on CH, especially considering the fact that there can
>>>be no such thing as continuum in the physical world.
>>
>>That's a non sequitur in at least 2 ways.
>>
>>1) CH is about cardinals and ordinals, not about the continuum.
> 
> Really. What does the continuum hypothesis say about the continuum?

The Pythagorean theorem is not about Pythagoras.
CH does not talk about the set of real numbers. It talks about its 
cardinality.

>>2) how can you say "there -can- be no continuum" in the physical
>>world? The continuum is a concept. There also can't be a 2 in the
>>physical world
> 
> 2 is a number. We observe many *copies* of this concept in the physical
> world, e.g. there are such things as binary quantum states. We observe
> no such thing as actual infinite divisibility in the physical world,

OK, you got me there. We can't observe an infinity (denumerable or 
not) in the physical world (not in limited time!). But that is sorta 
by definition.

> there is no such thing as a continuous space as the property of _any_
> subset of space-time. It simply does not exist, and furthermore in a
> mechanical world like ours it _cannot_ exist. 

You have not convinced me of that. I don't see how it would be 
possible to test the physical world sufficiently to convince anyone 
one way or the other.

> This is a deep
> metaphysical difference between an integer and a _real_ number.

I interesting that the side and diagonal of a square are 
incommensurable (no integer multipliers making them equal), and that 
there is no integer coefficient polynomial relation between the 
diameter and circumference of a circle. But I suspect you mean more 
than this.

Or did you mean there is a deep metaphysical difference between Z and R?

> One would have to forget the obsolete platonist way of talking about
> mathematics if one would like to understand the difference. There can
> be no such thing as "the truth" to the CH, I basically think
> Kronecker's comments on this problem were exactly true. This shows
> itself, but those who have some kind of sympathy to naive set theory,
> naive geometry and naive number theory want to hold on to these
> notions. It seems like an emotional reaction of some sort.

That is probably so. The emotional reaction I do know I had was with 
your statement "there can be no such thing as continuum in the 
physical world", which sounded just too (baselessly) authoritarian to 
pass up comment.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
2/2/2005 12:55:38 PM
Mitch Harris says...

>>>1) CH is about cardinals and ordinals, not about the continuum.

Um, yes it is. The continuum is another word for the set of real
numbers, and the continuum hypothesis is the hypothesis that the
continuum has cardinality omega_1.

--
Daryl McCullough
Ithaca, NY

0
2/2/2005 3:09:09 PM
Daryl McCullough wrote:
> Mitch Harris says...
> 
>>>>1) CH is about cardinals and ordinals, not about the continuum.
> 
> Um, yes it is. The continuum is another word for the set of real
> numbers, and the continuum hypothesis is the hypothesis that the
> continuum has cardinality omega_1.

OK. maybe I'm splitting hairs. here's another way of saying CH: the 
number of subsets of N is equal to the second smallest infinite cardinal.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
2/2/2005 3:54:15 PM
Mitch Harris wrote:
> That is probably so. The emotional reaction I do know I had was with
> your statement "there can be no such thing as continuum in the
> physical world", which sounded just too (baselessly) authoritarian to

> pass up comment.

I do not mean to be authoritative. That would be a Torkelism.

I think the evidence for a discrete world far outweighs the evidence
for a continuous world, which is basically non-existent.

If the world were continuous, then there might be a way to store a real
number as a physical property. However, all storage devices have to
rely on fundamental properties in the atomic world which are _all_
discrete, e.g. quantum physics.

There is also something called Heisenberg's uncertainty principle. Why
would I believe that something exists beneath the Planck scale, while
our physics tells us that you cannot physically subdivide the Planck
scale. There is no such thing, as far as I can tell, as to measure time
or space in fractions of the Planck scale. I would of course be
interested to know if there is a work that shows the Planck scale is
bogus!

The problem with that kind of a belief is its similarity to theological
"reasoning". There is something called "God" that is fundamentally
unobservable, but some people believe in its existence. Substitute God
with continuum. (That is I object also on metaphilosophical grounds)

The only "evidence" for a continuous world is classical and
relativistic physics cast in the  language of geometry which makes use
of real-valued numbers, that is they are no evidence. (How can a
"theoretical assumption" be an evidence?) If we take particle physics
seriously, which we should, we cannot say that they are equal ways of
describing the world.

Here, something interesting you might ask: but the wave function is
continuous right? Right. Does the wave function exist? I don't think
so. It is merely another theoretical instrument.
Regards,

--
Eray Ozkural

0
examachine (384)
2/2/2005 4:01:57 PM
 <examachine@gmail.com> wrote:
>
>I think the evidence for a discrete world far outweighs the evidence
>for a continuous world, 

what little I know of contemporary physics seems to agree with this, at 
certain very small scales.

>which is basically non-existent.

evidence for a continuous world non-existent? I couldn't say.

>If the world were continuous, then there might be a way to store a real
>number as a physical property. However, all storage devices have to
>rely on fundamental properties in the atomic world which are _all_
>discrete, e.g. quantum physics.

Have to? that's a bit unimaginative.


>There is also something called Heisenberg's uncertainty principle. Why
>would I believe that something exists beneath the Planck scale, while
>our physics tells us that you cannot physically subdivide the Planck
>scale. There is no such thing, as far as I can tell, as to measure time
>or space in fractions of the Planck scale. I would of course be
>interested to know if there is a work that shows the Planck scale is
>bogus!
>
>The problem with that kind of a belief is its similarity to theological
>"reasoning". There is something called "God" that is fundamentally
>unobservable, 

(this is quite another topic but some people claim very clear evidence
of such things)


>but some people believe in its existence. Substitute God
>with continuum. (That is I object also on metaphilosophical grounds)

OK. Substitute God with discrete. same problems (or same difficulties 
removed)


>The only "evidence" for a continuous world is classical and
>relativistic physics cast in the  language of geometry which makes use
>of real-valued numbers, that is they are no evidence. (How can a
>"theoretical assumption" be an evidence?) 

just as the theoretical structures of group theory that provide some of 
the language of quantum theory are not evidence. 


>If we take particle physics
>seriously, which we should, we cannot say that they are equal ways of
>describing the world.

one way might be more successful in describing the world.


>Here, something interesting you might ask: but the wave function is
>continuous right? Right. Does the wave function exist? I don't think
>so. It is merely another theoretical instrument.

(sorry to keep using the same example but) 2 is merely another theoretical 
instrument.

Mitch

0
harrisq (267)
2/2/2005 9:36:48 PM
examachine@gmail.com wrote:
| Mitch Harris wrote:
| > That is probably so. The emotional reaction I do know I had was
with
| > your statement "there can be no such thing as continuum in the
| > physical world", which sounded just too (baselessly) authoritarian
to
| > pass up comment.
|
| I do not mean to be authoritative. That would be a Torkelism.

Well, he doesn't generally sling around this kind of loose
claim the way you're doing. If you don't want to sound like
you're presenting yourself as an authority, stop using such
phrases as "just doesn't exist", "it _cannot_ exist", "there
is no such thing" without giving some solid reason to believe
that they're accurate. I think you're just being very glib.

| I think the evidence for a discrete world far outweighs the evidence
| for a continuous world, which is basically non-existent.

On the contrary, there's essentially no evidence that the
world is discrete. Really, there's not much that could
reasonably be called evidence in either direction.

If the world were discrete, one could hope to observe the
fact by examining it at a small enough scale. In principle,
then, one should be able to model it at that level. But none
of our best actually working models of the world is entirely
discrete.

The approach to quantum gravity known as "spin networks"
comes close, but still the state of a system is
a superposition of states, where the weights can vary
continuously. John Baez has pointed out that it's also
consistent to have both a model such as the spin network
model and a model in which the states are treated as having
continuous space. The model is discrete in some respects
and continuous in others.

There are problems that naturally arise for lots of ways
that you could attempt to produce a discrete model of
nature. Ones that treat space as a lattice, for example,
tend to predict the existence of "preferred" directions
in space, or a preferred state of rest, which we don't see.
Getting a discrete model to be relativistically invariant
is a bit of a challenge.

| If the world were continuous, then there might be a way to store a
real
| number as a physical property. However, all storage devices have to
| rely on fundamental properties in the atomic world which are _all_
| discrete, e.g. quantum physics.

Not unless you assume what you're setting out to prove. If
we record something on an analog device, there usually is no
reason to think that the value being stored is one of a
discrete set of possibilities. People read about how the
energy levels of a bound system are discrete and get the
idea that according to quantum mechanics all quantities are
discrete, but it's not so.

A lot of the arguments that are put forth in favor of nature
being discrete are really only arguments in favor of nature
being modelled by a separable space. In topology a space is
separable if it has a countable dense subset. By giving
enough discrete information one can (apparently) describe a
physical situation to any desired degree of accuracy. But
if so that merely means that the state is a point in some
separable space, not that there is some ultimate level of
precision on which there are discrete steps from one state
of the system to the next.

| There is also something called Heisenberg's uncertainty principle.
Why
| would I believe that something exists beneath the Planck scale, while
| our physics tells us that you cannot physically subdivide the Planck
| scale.

Where does it say that? The Planck length is simply a length
small enough that to model physics on that scale, quantum
gravity effects have to be taken into consideration.

| There is no such thing, as far as I can tell, as to measure time
| or space in fractions of the Planck scale. I would of course be
| interested to know if there is a work that shows the Planck scale is
| bogus!

It's not so much that it's bogus, as that you're interpreting
what it means in a simplistic way. The state of a small piece
of space may well be in a superposition, where the location or
identity of points is indeterminate in some way. It doesn't
follow that the way to think of them is as a finite or even as
a countable collection.

| The problem with that kind of a belief is its similarity to
theological
| "reasoning". There is something called "God" that is fundamentally
| unobservable, but some people believe in its existence. Substitute
God
| with continuum. (That is I object also on metaphilosophical grounds)

I don't think the analogy is a good one. The only obvious
common feature of the two ideas is that you don't like them.
You're treating belief that something (like space) is
continuous as if it were the positive belief in some exotic
entity.

But it's the idea that space is discrete that's the positive
claim, not yet observed or verified. If space is discrete,
we presumably will eventually have a theory that identifies
the individual bits of it and can tell us how large a gap
there is, experiments that exhibit the effects of there
being such granularity, and so on. This has not happened
yet. No, physics does not currently treat space as being
made up of little points sprinkled around at around a Planck
length apart; it just doesn't.

One common point of view of atheists is that the existence
of God is something that similarly might be experience (if
only it were actually true), but that so long as they have
no such experience, they will opt for the assumption that
there is none. Belief in the continuity of space is often
of the same form: show us a length scale on which space is
made up of discrete points and we'll agree it is, but until
then we won't assume that there is such a scale.

Of course it's actually worse than that, because it might be
that on some scale nature works to a *close approximation*
like a discrete model, but on even closer inspection the
discrete model also turns out to be inaccurate. So it's not
at all clear to me that there can be a final and convincing
answer to the question. (Actually, the same issue has been
pointed out in relation to theology; it's a little hard to
see how one would be entirely sure that any experience one
had had was actually of God.)

Continuous models are not necessarily any more complicated
than discrete ones, so Occam's razor and the like don't
guide us away from tentatively assuming space is continuous
either.

| The only "evidence" for a continuous world is classical and
| relativistic physics cast in the  language of geometry which makes
use
| of real-valued numbers, that is they are no evidence. (How can a
| "theoretical assumption" be an evidence?) If we take particle physics
| seriously, which we should, we cannot say that they are equal ways of
| describing the world.
|
| Here, something interesting you might ask: but the wave function is
| continuous right? Right. Does the wave function exist? I don't think
| so. It is merely another theoretical instrument.

It's fine to be circumspect about how well your models
correspond to reality, but you can't be arbitrarily
selective about it. For the time being, the best models
we have (simplest and covering the most phenomena) have
some kind of continuous element to them. Maybe you think
you can see beyond the veil presented by our models of
nature, and see that nature itself is one way or the
other, but I don't see how. Meanwhile, the best guide we
have to how nature is is given by those theories.

I don't consider the continuous nature of theories of
physics to be very strong evidence that nature is actually
continuous, but I think it's strong enough to refute the
glib assertions you've been making here, that a continuum
in nature just can't exist.

------------------------------------

In some sense I would say all of this is fairly irrelevant
to the meaningfulness of the continuum hypothesis. One way
that physics supports the meaningfulness of mathematics is
by deducing physical consequences with the use of certain
mathematical facts. You would be on much more solid ground
if you claimed merely that the continuum hypothesis will not
be needed to deduce observable consequences of any physical
theory. I think that's probably true.

On one axis, there's a gulf between Platonism and several
other philosophies of mathematics. Appealing to physics here
might sway some people toward thinking that some mathematical
question is or is not directly meaningful, but the Platonist
has already bitten the bullet in deciding that the meaningfulness
of a question doesn't depend upon there being any way even in
principle for us to answer it, either by proof, calculation,
or experiment.

Keith Ramsay

0
kramsay (59)
2/3/2005 8:12:15 AM
examachine@gmail.com wrote:
> 
> I think the evidence for a discrete world far outweighs the evidence
> for a continuous world

I think Keith put much better into words my complaints. But I think 
I'll also add some of my motivation. I don't think you (anyone) can 
determine experimentally whether "reality" is continuous or discrete. 
One can certainly experimentally corroborate that a discrete model 
works better in particular circumstances than a continuous one. But to 
say that reality is or is not one or the other is just way too 
narrowly focussed.

And now my appeal to authority (probably where I got this point of 
view originally). In Kant's Prolegomena to Any Future Metaphysics 
(text is online), he addresses -exactly- this issue. Sections 51-53, 
particularly 52c discuss some "either-or" antinomies (bounded vs 
infinite, simple vs composite (I take this to mean discrete or 
continuous)). And he concludes that both are false, without appealing 
to results of scientific experiment.

-- 
Mitch Harris
(remove q to reply)

0
harrisq (267)
2/3/2005 10:06:55 AM
examachine@gmail.com wrote:
> I think the evidence for a discrete world far outweighs the evidence
> for a continuous world, which is basically non-existent.

Then why is it that physicists use real numbers and continuous functions 
more than anyone else? They *invented* those things! (or discovered, or 
whatever...)

> If the world were continuous, then there might be a way to store a real
> number as a physical property. However, all storage devices have to
> rely on fundamental properties in the atomic world which are _all_
> discrete, e.g. quantum physics.

Quantum physics is not as discrete as you seem to think. I have even seen 
arguments that quantum theory cannot be correct because it is not discrete.

Consider a single qbit (e.g. the spin of an electron). In a sense, this is 
as discrete as any quantum system can ever be. If you use it to store 
classical information, its capacity is just one bit.

However, its behavior cannot be perfectly reproduced by a discrete system 
of *any* finite size. It can be approximated to arbitrary precision, but so 
can a continuous system. To even *approximate* a collection of interacting 
spins seems to require a discrete system that grows exponentially as the 
number of spins increases.

Also, fully discrete approximations to quantum phenomena are very contrived 
and complicated - they are not good as physical theories.

Quantum physics requires a continuum, even though it may be discrete in the 
sense of not allowing a real number to be stored exactly (I don't know for 
sure if it does or not).

> Why
> would I believe that something exists beneath the Planck scale, while
> our physics tells us that you cannot physically subdivide the Planck
> scale. There is no such thing, as far as I can tell, as to measure time
> or space in fractions of the Planck scale.

I have never heard such a thing from an actual physicist (from science 
journalists yes), and I follow work on quantum gravity. I know that there 
is *no* consistent physical theory with an indivisible unit of space or 
time. There are discrete theories of gravity (none good enough to test), 
but they are not discrete in *that* way.

> The only "evidence" for a continuous world is classical and
> relativistic physics cast in the  language of geometry which makes use
> of real-valued numbers, that is they are no evidence. (How can a
> "theoretical assumption" be an evidence?) If we take particle physics
> seriously, which we should, we cannot say that they are equal ways of
> describing the world.

So we should take quantum theory, but not general relativity, seriously? I 
certainly take them *both* seriously!

> but the wave function is
> continuous right? Right. Does the wave function exist? I don't think
> so. It is merely another theoretical instrument.

But the discrete features of quantum theory are just "theoretical 
instruments" too. If you accept quantum theory, as anyone who is serious 
about physics must, you have to accept it all. There are alternate 
formulations of quantum mechanics that have no "wave function", but they 
all contain *some* continuous parts.

There are also alternate "interpretations" of quantum theory (just as there 
are alternate interpretations of probability theory). They are irrelevant 
to the question at hand.

Ralph Hartley
0
hartley (156)
2/3/2005 1:39:08 PM
examachine@gmail.com wrote:
> Mitch Harris wrote:
> 
>>That is probably so. The emotional reaction I do know I had was with
>>your statement "there can be no such thing as continuum in the
>>physical world", which sounded just too (baselessly) authoritarian to
> 
> 
>>pass up comment.
> 
> 
> I do not mean to be authoritative. That would be a Torkelism.
> 
> I think the evidence for a discrete world far outweighs the evidence
> for a continuous world, which is basically non-existent.

I don't see why it matters, either way.

If it was proven tomorrow by physicists that space was continuous
it wouldn't invalidate discrete mathematics, would it?  OTOH, if
was proven tomorrow that space was discrete it wouldn't
invalidate any mathematics based on the continuum.

I think the whole notion of getting rid of the reals
is just plain kooky.

Since you only seem to be interested in applications of mathematics,
what competing alternatives are there to the current theory of
calculus that aren't based on the existence of sets with cardinality
greater than aleph_0?

Are those theories easier for engineers and such to learn and
manipulate?

If a competing theory produces essentially the same results as
the standard theory of calculus, then why prefer the other
theory?

If any of the alternative theories of calculus are superior, then
why haven't they caught on in the last 400 years?

[...]

-- 
Replace Roman numerals with digits to reply by email
0
2/3/2005 6:07:11 PM
In article <1107418335.324750.294210@o13g2000cwo.googlegroups.com>,
Keith Ramsay <kramsay@aol.com> wrote:

>examachine@gmail.com wrote:

>| I think the evidence for a discrete world far outweighs the evidence
>| for a continuous world, which is basically non-existent.

>On the contrary, there's essentially no evidence that the
>world is discrete. Really, there's not much that could
>reasonably be called evidence in either direction.

Hi, Keith.

Indeed, there's not a shred of experimental evidence that 
"the world is discrete".  If you take quantum theory seriously,
it's natural to guess it applies even to the geometry of spacetime,
and this would mean that you can't simultaneously measure everything 
about the geometry of spacetime with arbitrary precision.  But, that's
not yet "discreteness".   Quantum theory allows for lots of options.

For example, in ordinary quantum mechanics you can't measure the 
position and velocity of a particle both at the same time with 
arbitrarily good precision, but there's nothing "discrete" going 
on here.  You can measure either the position or velocity with as
much precision as you like, and they don't come in discrete steps.

There are other quantum systems, like the energy levels of an atom, 
that show a kind of discreteness - though not the naive discreteness 
of evenly spaced steps.

And while a bunch of people including myself have worked on theories 
where area and volume are "discrete" in about the same way as the 
energy levels of an atom:

Loop Quantum Gravity
http://math.ucr.edu/home/baez/acm/

these are still theories, not "evidence" of discreteness.  And, they 
are highly controversial theories!

>If the world were discrete, one could hope to observe the
>fact by examining it at a small enough scale. In principle,
>then, one should be able to model it at that level. But none
>of our best actually working models of the world is entirely
>discrete.

Right.

>The approach to quantum gravity known as "spin networks"
>comes close, but still the state of a system is
>a superposition of states, where the weights can vary
>continuously. John Baez has pointed out that it's also
>consistent to have both a model such as the spin network
>model and a model in which the states are treated as having
>continuous space. The model is discrete in some respects
>and continuous in others.

Right.  And, the spin network theory of quantum gravity has not
received any experimental confirmation thus far.

>| There is also something called Heisenberg's uncertainty principle.  Why
>| would I believe that something exists beneath the Planck scale, while
>| our physics tells us that you cannot physically subdivide the Planck
>| scale.

>Where does it say that? The Planck length is simply a length
>small enough that to model physics on that scale, quantum
>gravity effects have to be taken into consideration.

Right.  And in fact, even this is just a guess.  To be very clear,
we should admit that the Planck length is the simplest quantity
with units of length that we can cook up from the speed of light (c),
Newton's gravitational constant (G), and Planck's constant (hbar).  
It's about 1.6 x 10^{-35} meters.  

By dimensional analysis, we can *guess* that if quantum gravity
effects become important at some length scale, it's around the
Planck length.

But, this guess assumes that no other physical quantities are
important in determining this length scale!  E.g., not the mass
of the electron, or anything else like that.  

So, this guess could easily be wrong.  And, it's important to
remember that we don't have any direct evidence for what happens
at the Planck scale, or even length scales much bigger than this.

The diameter of a proton is about 10^{-15} meters.   We've done 
experiments that probe much shorter distance scales, but they're
still vastly larger than the Planck length.

I always forget how short are the distance scales we've probed so
far... let me work it out.  Cheating, I'll start by looking in T. D.
Lee's book on particle physics, in the chapter "Order-of-Magnitude
Estimates"... let's see... good!  He says the electron mass is .51
MeV, and that this corresponds to (4 x 10^{-11} cm)^{-1}, where
he's using c and hbar to convert energies to inverse lengths.  
So, doing particle collision experiments at an energy of .51 MeV 
we can probe distance scales of about 4 x 10^{-13} meters.   Or,
roughly, 1 Mev corresponds to 10^{-13} meters.  That's what I should
remember!  Anyway, the best accelerator in the world is still LEP
(until LHC comes online), and that reached energies of about 113 GeV.
So, roughly 100 GeV, or 10^5 MeV - so distance scales of about 10^{-18}
meters.

So, unless I made a stupid mistake, we can currently probe distances
about 1/1000th the size of a proton, and we haven't seen any trace of
spacetime discreteness... 

.... but these length scales are still about 10^{17} times as big as 
the Planck scale!   

So, when we are speculating about what happens at the Planck length,
we are extrapolating our ideas on physics to distance scales that are
100,000,000,000,000,000 smaller than anything we have experimental 
access to, and hoping that nothing really unexpected happens at these
shorter distances!  

This is a wild extrapolation.  Physicists indulge in it mainly because 
it's more fun to think about what physics would be like at these distance
scales based on what we know, than to throw up our hands in despair
give up.  

So, I would not say "our physics tells us that you cannot physically 
subdivide the Planck scale". 

>In some sense I would say all of this is fairly irrelevant
>to the meaningfulness of the continuum hypothesis. 

Wow!  Is that what you guys were talking about?   I would never
have guessed.  Yeah - fairly irrelevant!!!

By the way, I hope someone pointed out that the capitalized 
statement in the subject of this thread is false, and provably
false in, say, Peano arithmetic.






0
baez (1)
2/3/2005 8:52:47 PM
John Baez wrote:
> In article <1107418335.324750.294210@o13g2000cwo.googlegroups.com>,
> Keith Ramsay <kramsay@aol.com> wrote:
>
> >examachine@gmail.com wrote:
>
> >| I think the evidence for a discrete world far outweighs the
evidence
> >| for a continuous world, which is basically non-existent.
>
> >On the contrary, there's essentially no evidence that the
> >world is discrete. Really, there's not much that could
> >reasonably be called evidence in either direction.

I disagree, there is evidence that allready exists that suggests that
the universe is neither continuous or discrete, it is fractal. Besides
just looking around at all the fractals you can see in Nature with ones
own eye, the evidence comes from physics. Specifically, in QFT, to make
sense of a number of Feynmann diagrams you have to use Renormalization.


If you look at the Renormalization formulas, they look surprisingly
"like" the Hausdorff dimension formula. This is not a mistake as far as
I can tell. QFT got the idea from Condensed Matter physics, and
condensed matter physicists got the idea from the mathematicians. The
QFT people even use the terms from dimension theory, "scale", "flow",
etc...

I think though because where the idea for "Renormalization" came from
became lost (dimension theory), or because a lot of physicists believe
in Democritus like thinking, the idea that the world might be fractal
all the way down has never been considered seriously. There is even a
term for what the Renormalization procedure does that bypasses the idea
that the procedure might be usefull because of dimension theory:
cutoff. The procedure cuts off our ignorance of what happens in a realm
with higher energy. Maybe what the procedure really does is cut off our
ignorance that the world is discrete or continuous, because it is
fractal!
                                       -- NPC

note: I do agree that Renormalization also gives us a cutoff, but that
would have just complicated the argument.

0
qmagick (1)
2/3/2005 11:31:00 PM
qmagick@yahoo.com wrote
> a number of Feynmann diagrams ...

5 points.

(Hint: see:
<http://math.ucr.edu/home/baez/crackpot.html>)

--- Jeff


0
ketland (18)
2/4/2005 12:22:12 AM
"John Baez" <baez@math-cl-n03.math.ucr.edu> wrote in message
news:ctu2uv$pa$1@glue.ucr.edu...
| In article <1107418335.324750.294210@o13g2000cwo.googlegroups.com>,
| Keith Ramsay <kramsay@aol.com> wrote:
|
| >examachine@gmail.com wrote:
|
| >| I think the evidence for a discrete world far outweighs the
evidence
| >| for a continuous world, which is basically non-existent.
|
| >On the contrary, there's essentially no evidence that the
| >world is discrete. Really, there's not much that could
| >reasonably be called evidence in either direction.
|
| Hi, Keith.
|
| Indeed, there's not a shred of experimental evidence that
| "the world is discrete".  If you take quantum theory seriously,
| it's natural to guess it applies even to the geometry of spacetime,
| and this would mean that you can't simultaneously measure everything
| about the geometry of spacetime with arbitrary precision.  But, that's
| not yet "discreteness".   Quantum theory allows for lots of options.
|
| For example, in ordinary quantum mechanics you can't measure the
| position and velocity of a particle both at the same time with
| arbitrarily good precision, but there's nothing "discrete" going
| on here.  You can measure either the position or velocity with as
| much precision as you like, and they don't come in discrete steps.
|
| There are other quantum systems, like the energy levels of an atom,
| that show a kind of discreteness - though not the naive discreteness
| of evenly spaced steps.
|
| And while a bunch of people including myself have worked on theories
| where area and volume are "discrete" in about the same way as the
| energy levels of an atom:
|
| Loop Quantum Gravity
| http://math.ucr.edu/home/baez/acm/
|
| these are still theories, not "evidence" of discreteness.  And, they
| are highly controversial theories!
|
| >If the world were discrete, one could hope to observe the
| >fact by examining it at a small enough scale. In principle,
| >then, one should be able to model it at that level. But none
| >of our best actually working models of the world is entirely
| >discrete.
|
| Right.
|
| >The approach to quantum gravity known as "spin networks"
| >comes close, but still the state of a system is
| >a superposition of states, where the weights can vary
| >continuously. John Baez has pointed out that it's also
| >consistent to have both a model such as the spin network
| >model and a model in which the states are treated as having
| >continuous space. The model is discrete in some respects
| >and continuous in others.
|
| Right.  And, the spin network theory of quantum gravity has not
| received any experimental confirmation thus far.
|
| >| There is also something called Heisenberg's uncertainty principle.
Why
| >| would I believe that something exists beneath the Planck scale,
while
| >| our physics tells us that you cannot physically subdivide the
Planck
| >| scale.
|
| >Where does it say that? The Planck length is simply a length
| >small enough that to model physics on that scale, quantum
| >gravity effects have to be taken into consideration.
|
| Right.  And in fact, even this is just a guess.  To be very clear,
| we should admit that the Planck length is the simplest quantity
| with units of length that we can cook up from the speed of light (c),
| Newton's gravitational constant (G), and Planck's constant (hbar).
| It's about 1.6 x 10^{-35} meters.
|
| By dimensional analysis, we can *guess* that if quantum gravity
| effects become important at some length scale, it's around the
| Planck length.
|
| But, this guess assumes that no other physical quantities are
| important in determining this length scale!  E.g., not the mass
| of the electron, or anything else like that.
|
| So, this guess could easily be wrong.  And, it's important to
| remember that we don't have any direct evidence for what happens
| at the Planck scale, or even length scales much bigger than this.
|
| The diameter of a proton is about 10^{-15} meters.   We've done
| experiments that probe much shorter distance scales, but they're
| still vastly larger than the Planck length.
|
| I always forget how short are the distance scales we've probed so
| far... let me work it out.  Cheating, I'll start by looking in T. D.
| Lee's book on particle physics, in the chapter "Order-of-Magnitude
| Estimates"... let's see... good!  He says the electron mass is .51
| MeV, and that this corresponds to (4 x 10^{-11} cm)^{-1}, where
| he's using c and hbar to convert energies to inverse lengths.
| So, doing particle collision experiments at an energy of .51 MeV
| we can probe distance scales of about 4 x 10^{-13} meters.   Or,
| roughly, 1 Mev corresponds to 10^{-13} meters.  That's what I should
| remember!  Anyway, the best accelerator in the world is still LEP
| (until LHC comes online), and that reached energies of about 113 GeV.
| So, roughly 100 GeV, or 10^5 MeV - so distance scales of about
10^{-18}
| meters.
|
| So, unless I made a stupid mistake, we can currently probe distances
| about 1/1000th the size of a proton, and we haven't seen any trace of
| spacetime discreteness...

Is a proton part of spacetime?  If so, then we have seen a trace of
spacetime discretness.

FrediFizzx

0
2/4/2005 2:26:18 AM
Torkel Franzen wrote:
> poopdeville@gmail.com writes:
>
> > A set of syntactic rules is more-or-less a set of transformations
> > converting one un-interpreted symbol into another.  In this context
we
> > might consider a particular formalization of the FOL and a set of
> > axioms for a language as a set of syntactic rules.  A set of
syntactic
> > rules, then, picks out a unique interpretation if there is a unique
> > model of the syntactic rules as explained here.
>
> > A simple example requires that we consider syntactic rules over
SOL,
> > but this is inessential to my point:  all complete ordered fields
are
> > isomorphic.
>
>   They are indeed, but what does this have to do with syntactic
rules?
> That all complete ordered fields are isomorphic is a set-theoretical
> theorem. It is not a statement that there is a unique model of any
> "syntactic rules". Indeed there is no standard notion in logic of a
> model of a set of syntactic rules in your sense of a set of
> "transformations converting one uninterpreted symbol into
> another".

The rules of inference and axioms of a theory satisfy my
quasi-definition of a set of "syntactic rules."  Indeed, a proof is a
sequence of transformations on uninterpreted symbols.  The "legal"
transformations are specified by the rules of inference one is allowed
to use and the axioms for the theory.

>That all complete ordered fields are isomorphic means that
> all models of a certain second order theory, that is, all structures
> in which the axioms of the theory are *true*, are isomorphic. How
> would you consider the axioms of the theory as "transformations
> converting..." etc?

What I was trying to get at is that when a mathematician talks about
real numbers, there is no ambiguity with respect to what he is talking
about since the axioms of the theory of real analysis are categorical.
No matter what his "private interpretation" is, as long as it satisfies
the axioms, he can communicate everything relevant.

Contrast this with group or set theory.  A group theorist might have a
"private interpretation" of a group.  It might, for the sake of
discussion, be equivalent to a particular group, say the cyclic group
of order 7.  When trying to prove things, he has to (1) make sure his
proof doesn't depend on properties specific to Z_7 (such as, "has
exactly 7 elements") or (2) state that he is working *specifically*
with Z_7, either by mentioning it in English (or German, or some other
natural language), which is the most likely case, or by adding an axiom
Z_7 to the group axioms so that GA + Z_7 picks out Z_7.

Set theory is similar in that we all probably have a particular model
in mind on which we base our intuitions.  In your "private
interpretation," AC might be true, and it might be false with respect
to mine. The whole point of using a formal language like ZF is to
divorce our "private interpretations" from our "public language."

'cid 'ooh

(Sorry for the late reply, I've been pretty busy)

0
poopdeville (133)
2/4/2005 3:48:05 AM
Sorry for the late reply.  I've been really busy.

Jeffrey Ketland wrote:
> poopdeville@gmail.com wrote in message
> >> If "syntactic rules demonstrably don't pick out a unique
> >interpretation",
> >> then can you explain why this implies "we need to be careful when
> >talking
> >> about truth"? (It is consistent with the premise that the
> >interpretation is
> >> "picked out" some other way. How do you know?)
> >
> >Because if we aren't careful, we'll have people with cavalier
attitudes
> >and ideologies writing nonsense in books and articles.
>
> I'm still finding your philosophical view quite puzzling.
> It seems to have the following overall reductionist structure:
>
> (i) there are notions D1, D2, D3, ..., that you consider determinate.
> (ii) there are notions M1, M2, M3, ... that you find meaningless,
> indeterminate or model-relative (unless they can be reduced to kosher
> notions in (i)).
>
> Then you classify these as follows
>
>     Determinate notions              Meaningless, indeterminate
notions
>     "syntactic rule",                         "set"
>     "formula",                                   "truth simpliciter"
>      "model",                                    "natural number"
>     "true-in-an-interpretation".
>
> Your arguments (correct me if I'm wrong!) for this classification go
roughly
> as follows:
>
> 1. The notion of "set" is indeterminate (or model-relative). This is
because
> the "syntactic rules" (a notion you accept as kosher) for ZF don't
"pick out
> a unique interpretation".
> 2. The notion of "truth simpliciter" is meaningless. This is because
the
> *only way* that truth simpliciter could be a kosher notion would be
via
> reductive analysis: i.e., if the "syntactic rules" of the language
"picked
> out a unique interpretation", then truth simpliciter would be truth
in this
> unique model.
> 3. The notion of "natural number" is indeterminate (or
model-relative). This
> is because there are non-standard models of the "syntactic rules" of
Peano
> arithmetic.
>
> Is that it?

Yes.  Though 1 and 3 are intances of the same phenomenon.

The essential point is that we each have intuitive notions of what
numbers are.  And we've agreed to talk about them using the rules of
inference of the FOL and the Peano axioms (these are the "syntactic
rules" I speak of.)  The language we speak admits *very* different
interpretations, as Godel showed.  Because of this, you and I cannot be
sure we have to same (mental or logical) model in mind when we speak
about numbers.  So we each have two options:  we prove theorems that
are true in each model of PA, or we find a way to axiomatize our
intuitive model and prove theorems about *that* -- this may even be
impossible.  If we persue the third option -- talk of truth and numbers
as if they existed in a vacuum or plenum or some other metaphysical
construct, we fail to communicate anything at all.

> (And, though it's not too important, you think that people who don't
agree
> with your philosophical views are not "careful" and have "cavalier
attitudes
> and ideologies" and they "write nonsense". Kurt Goedel, for example,
"wrote
> nonsense". Similarly, Hugh Woodin and others.)

I've not read much Woodin, but Godel did write a lot of philosophical
nonsense.  His technical work was excellent, but his realist ideology
blocked him from the philosophical insight to which his technical work
could have lead.

'cid 'ooh

0
poopdeville (133)
2/4/2005 4:00:54 AM
Sorry for the late reply.

tchow@lsa.umich.edu wrote:
>
> >Yes!  We need to know what symbols are, and what (the relevant)
> >syntactic rules are to do number theory.  But the syntactic rules
> >demonstrably don't pick out a unique interpretation, which is
exactly
> >why we need to be careful when talking about truth.
>
> To respond to this let me skip forward a bit to:
>
> >If you can tell me what a set is, outside of a "collection" (the
> >more-or-less personal "mental picture") or "an element of a
collection
> >which satisfies the membership relation" (the public notion that
> >doesn't pick out one of these mental pictures), that'd be great.
:-)
>
> I'll tell you what a set is if you can tell me what a symbol is.

Glyphs on paper, words, sense data, representations of objects that
must be interpreted.  Outside of that, I don't know what they are, but
I know we have much more immediate access to them than the things they
represent.  (This is the point where you can justifiably accuse me of
being an extreme skeptic)

>
> By the way, historically, symbols were often thought of as being less
well
> understood than integers.  The whole notion of Goedel numbering is,
in a
> sense, a symptom of the feeling that we know how to deal with
integers, so
> to convince ourselves that we know how to deal with symbols and
strings,
> let's convert symbols and strings to integers, so that we *really*
know
> what we're talking about.

This is interesting.

>
> But that's an aside.  Let me take your argument about unique
interpretations
> and turn it against you.  Logic, like any other branch of
mathematics, can
> be formalized and studied.  In particular, we can develop a
first-order
> theory of syntax, which captures discourse about symbols and strings
and
> so forth.  When we do this, we discover that sentences about symbols,
> strings, rules, and so forth admit nonstandard interpretations.  We
might
> *think* we know what we mean by a proof of finite length, but lo and
behold,
> nonstandard models of axioms for syntax show us that "finite length"
proofs
> could be "nonstandardly finite," which is what most people would call
> infinite.  Does this shake your belief that you know what symbols and
proofs
> are?

No.  But then again, I was already familiar with logics in which some
theorems require infinite tableux to prove.  And other various
non-classical logics, such as intuitionist logics.  This only helps
make my point for me -- unless we specify in which logic we're working
with, we're not going to be able to understand each other (OK, so FOL
is the de facto standard.  I have no quarrel with that.  It's up to the
intuitionist to mention that he's working with a non-classical logic)

>
> >> The set of axioms for a group does not mimic the *discourse* of
group
> >> theorists.
> [...]
> >I had not thought of this distinction.
>
> Let me elaborate a bit because I think this exercise can be helpful.
> Suppose we try to express "every finite simple group is X" in the
> first-order language of groups.  (I'm being lazy and not saying what
> X is, but as you'll see, we won't need to know anything about X other
> than that it's some property that groups might have.)  As a first
step
> towards formalization we might try to rephrase:
>
>   For all groups G, if G is finite and G is simple, then G is X.
>
> The problem is that we're quantifying over *groups* here ("for all
> groups G"), and in the first-order language of groups we can only
> quantify over *group elements*.  (That's what "first-order" *means*.)
>
> Let's not give up just yet, though.  If we could find a sentence
> "FiniteSimple" in the first-order language of groups such that
> "GroupAxioms & FiniteSimple" is satisfied by all and only the finite
> simple groups, then
>
>    GroupAxioms & FiniteSimple  ->  X
>
> would be a tolerably good formalization.  The problem, though, is
> the basic theorem of mathematical logic that any first-order sentence
> that is satisfiable by arbitrarily large finite structures is also
> satisfiable by some infinite structures.  So we can't write down a
> sentence that captures "finite" in the sense we want.
>
> Problems also arise with "simple" because this means that there are
no
> nontrivial normal subgroups.  A subgroup is a *subset* of the group,
> and again we can't talk directly about subsets in a first-order
> language, only about group elements.
>
> These limitations illustrate that when people study the first-order
> language of groups, they're *not* trying to mimic a large fraction of
> group-theoretic discourse; they're interested specifically in
*first-order
> properties of groups*, which is a very limited fraction of the
properties
> of groups that group theorists in general are interested in.
>
> In contrast, when people study ZFC, they are often investigating the
> foundations of mathematics as a whole, and it's important that most
of
> mathematical discourse can be mimicked more-or-less directly in the
> language.  It's not that we're interested in---for lack of a better
> term---"universes" and we're trying to isolate the first-order
properties
> of universes for special scrutiny.  We're actually trying to find
formal
> counterparts for all the kinds of statements that mathematicians
make.
>
> The two projects are quite different and the fact that the same tool
> (first-order logic) is being used in both cases does not mean that
> direct parallels between ZFC and the axioms for a group always make
> sense.

Agreed.  Thanks for the great explanation.  

'cid 'ooh

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