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Godel has no idea what truth is so incompleteness...

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Newberry...

Posted: Wed Nov 04, 2009 7:00 pm

Guest

[quote]
It is the notion that certain formulae are
true but not provable that makes the theorem interesting.

You are welcome to believe that if you want. I did not comment on
which version of the theorem is *interesting*; that is a subjective
matter. I observed that there *is* a syntactical version.
[/quote]
First of all the only reason we prove theorems is that they are
interesting.

Secondly, Goedel's sentence is either provable or unprovable. In the
first case it is true that it is provable and false that it is
unprovable. If it is unprovable it is very hard to avoid the
conclusion that Goedel's sentence is true. It is hard to leave the
truth out.

Thirdly, you cannot have logic without the notion of truth.

Marshall...

Posted: Wed Nov 04, 2009 9:24 pm

Guest

On Nov 4, 10:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]
Do you really think the notion of natural numbers is a syntactical notion?
[/quote]
Do you think a syntactic treatment of the natural numbers is
impossible?

Marshall

Nam Nguyen...

Posted: Thu Nov 05, 2009 1:14 am

Guest

Rupert wrote:
[quote]On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware.
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?

Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:

- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.

- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.

See. No notions such as "natural numbers", "arithmetic truth", etc...

Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...

Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?

In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?

We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.
[/quote]
But that's where you're wrong! You said before (right above):

[quote]Omega-consistency is a syntactical notion.
[/quote]
Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?

Do you really think the notion of natural numbers is a syntactical notion?

Nam Nguyen...

Posted: Thu Nov 05, 2009 1:32 am

Guest

Rupert wrote:

[quote]I observed that there *is* a syntactical version.
[/quote]
Can you tell us all again who invented the truth version and who
the syntactical version?

Nam Nguyen...

Posted: Thu Nov 05, 2009 2:30 am

Guest

Marshall wrote:
[quote]On Nov 4, 10:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Do you really think the notion of natural numbers is a syntactical notion?

Do you think a syntactic treatment of the natural numbers is
impossible?
[/quote]
Of course it's impossible: you can't never demonstrate the syntactical
treatment - which is simply just a formal system as strong as Q - be
consistent, purely by syntactical means (axioms and rules of inference!

(Naturally, an in consistent system is also as strong as Q!).

Nam Nguyen...

Posted: Thu Nov 05, 2009 2:33 am

Guest

Nam Nguyen...

Posted: Thu Nov 05, 2009 2:58 am

Guest

Nam Nguyen wrote:
[quote]Marshall wrote:
On Nov 4, 10:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Do you really think the notion of natural numbers is a syntactical
notion?

Do you think a syntactic treatment of the natural numbers is
impossible?

Of course it's impossible: you can never demonstrate the syntactical
treatment - which is simply just a formal system as strong as Q - be
consistent, purely by syntactical means (axioms and rules of inference!

(Naturally, an in consistent system is also as strong as Q!).

[/quote]
In other words, from the syntactical point of view, the concept
of the natural numbers is just the concept of a consistent Q.

And there, is our problem: there's no syntactically-based decision
procedure to let us know which side of (in)consistency Q is in.

If we get angry with the strictness of the syntactical paradigm and
gamble with the intuitive-and-loose concept of natural numbers as
arithmetic truths, then we'd quickly get into a foundational problem:

(1) For any concept as strong as the natural numbers, there's an
arithmetic truth that it is impossible to know its truth value.

So there we have it: stay with the strictness of syntactical-ism and
we have to admit some consistency we *can't know* how to prove syntactically;
or venture into the looseness of intuition and face-to-face with
with a truth we *can't know* the truth value!

Either way we've lost and infinity has won. Why don't we be honest and
admit our limitation and move on to a better reasoning framework?

Jesse F. Hughes...

Posted: Thu Nov 05, 2009 7:31 am

Guest

Newberry <newberryxy at (no spam) gmail.com> writes:

[quote]
It is the notion that certain formulae are
true but not provable that makes the theorem interesting.

You are welcome to believe that if you want. I did not comment on
which version of the theorem is *interesting*; that is a subjective
matter. I observed that there *is* a syntactical version.

First of all the only reason we prove theorems is that they are
interesting.

Secondly, Goedel's sentence is either provable or unprovable. In the
first case it is true that it is provable and false that it is
unprovable. If it is unprovable it is very hard to avoid the
conclusion that Goedel's sentence is true. It is hard to leave the
truth out.
[/quote]
I think you mean decidable in the paragraph above, not provable. If
Goedel's sentence is not decidable, then it must be true --- that
seems right to me (though it is decidable, because it is provable).

[quote]Thirdly, you cannot have logic without the notion of truth.
[/quote]
It all depends on what you mean by logic, but as far as I can tell,
the syntax for FOL long preceded the semantics. Sure seems like they
had a logic back then.

--
"Your people are about denial. Dreams versus reality. TELLING
yourselves you are great. Telling yourselves you are brilliant.
Telling yourselves you understand mathematics."
--James S. Harris: So obvious that it's kind of sad.

Jesse F. Hughes...

Posted: Thu Nov 05, 2009 9:07 am

Guest

"Jesse F. Hughes" <jesse at (no spam) phiwumbda.org> writes:

[quote]Newberry <newberryxy at (no spam) gmail.com> writes:

It is the notion that certain formulae are
true but not provable that makes the theorem interesting.

You are welcome to believe that if you want. I did not comment on
which version of the theorem is *interesting*; that is a subjective
matter. I observed that there *is* a syntactical version.

First of all the only reason we prove theorems is that they are
interesting.

Secondly, Goedel's sentence is either provable or unprovable. In the
first case it is true that it is provable and false that it is
unprovable. If it is unprovable it is very hard to avoid the
conclusion that Goedel's sentence is true. It is hard to leave the
truth out.

I think you mean decidable in the paragraph above, not provable. If
Goedel's sentence is not decidable, then it must be true --- that
seems right to me (though it is decidable, because it is provable).
[/quote]

In retrospect, I'm not sure whether that's correct or not, since I'm
not sure what you're calling Goedel's sentence. If you mean the
sentence

(Ex)(~Prov(x) & ~Prov(NOT(x)))

where NOT(x) is the Goedel number of the sentence formed by negating
the sentence represented by x, then this sentence is apparently true
if it is not decidable. But, as I recall, you usually have a
different sentence in mind when you refer to "Goedel's sentence".

--
Jesse F. Hughes
"I have put all the information that you need at [a Yahoo! group] where
you'll notice a significantly better signal to noise ratio, as I'm
just about the only person posting." -- James S. Harris on noise

Charlie-Boo...

Posted: Thu Nov 05, 2009 11:08 am

Guest

On Nov 2, 2:24 am, byron <spermato... at (no spam) yahoo.com> wrote:
[quote]the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
[/quote]
Any system for representing sets that can represent the complement of
any set that it can represent will do. For example, "It is red."
defines the set of red things, and "It is red . . . not!" represents
its complement. The formal wff whose truth defines the set can also
be negated with a not symbol. Any method for creating sets of wffs
will do if its inverse is closed over complement: if a set has a
representation then its complement does.

This then must not equal proof because while we can represent the set
of provable sentences (Godel numbers) we cannot represent the set of
unprovable sentences. Proof does not equal truth or any other
negation-complete system of defining sets of wffs.

C-B

Do you agree that such a characterization of wffs exists? Then that
gives a description whose set is not represented by proof.

[quote]http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
[quote]Gdel's first incompleteness theorem, states that:

Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for
any consistent, effectively generated formal theory that proves
certain basic arithmetic [b]truths[/b], there is an arithmetical
statement that is[b] true,[/b][1] but not provable in the theory. [/
quote]

but Godel had no idea of what truth is as peter smith of cambridge
admitts

thus his incompleteness theorems is meaningless rubbish

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

[quote]mathematician have so much invested in godels incompleteness
theorem
much maths is reliant on it but at the time godel wrote his theorem he
had
no idea of what truth was as peter smith the Cambridge expert on Godel
admitts

http://groups.google.com/group/sci.logic/browse_thread/thread/ebde70b...
de566912ee69f0a8?lnk=gst&q=G%C3%B6del+didn%27t+rely+on+the+notion+PETER
+smith#de 566912ee69f0a8

Quote:
[b]Gdel didn't rely on the notion
of truth[/b]

but truth is central to his theorem
as peter smith kindly tellls us

http://assets.cambridge.org/97805218...40_excerpt.pdf
Quote:
Godel did is find a general method that enabled him to take any theory
T strong enough to capture a modest amount of basic arithmetic and
construct a corresponding arithmetical sentence GT which encodes the
claim The sentence GT itself is unprovable in theory T. So G T is
[b]true[/b] if and only if T cant prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
[b]true[/b]-but-unprovable,
[/quote]

thus godels incompleteness theorem is about true statements which cant
be proven
but godel cant tell us what makes a mathematical statement true

thus his theorem is meaningless

Abram Demski
notes

http://omgili.com/newsgroups/sci/logic/5f72458d70b444f3fa127507e37747...

With no working definition of truth, Godels proof cannot be taken
through the last step which converts the formal result about sentences
that can and can't be proven into one about truth and
incompleteness.The formal result still holds, though; it is just of
questionable interest. Right?

in fact his theorem is meaningless as he cant tell us what truth is[/quote]

Charlie-Boo...

Posted: Thu Nov 05, 2009 11:11 am

Guest

On Nov 5, 4:08 pm, Charlie-Boo <shymath... at (no spam) gmail.com> wrote:
[quote]On Nov 2, 2:24 am, byron <spermato... at (no spam) yahoo.com> wrote:

the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
[/quote]
Then Colin Dean needs to study a book on truth and logic. I
understand truth and Godel's theorems mean a lot to me.

C-B

[quote]Any system for representing sets that can represent the complement of
any set that it can represent will do. For example, "It is red."
defines the set of red things, and "It is red . . . not!" represents
its complement. The formal wff whose truth defines the set can also
be negated with a not symbol. Any method for creating sets of wffs
will do if its inverse is closed over complement: if a set has a
representation then its complement does.

This then must not equal proof because while we can represent the set
of provable sentences (Godel numbers) we cannot represent the set of
unprovable sentences. Proof does not equal truth or any other
negation-complete system of defining sets of wffs.

C-B

Do you agree that such a characterization of wffs exists? Then that
gives a description whose set is not represented by proof.

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
[quote]Gdel's first incompleteness theorem, states that:

Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for
any consistent, effectively generated formal theory that proves
certain basic arithmetic [b]truths[/b], there is an arithmetical
statement that is[b] true,[/b][1] but not provable in the theory. [/
quote]

but Godel had no idea of what truth is as peter smith of cambridge
admitts

thus his incompleteness theorems is meaningless rubbish

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

[quote]mathematician have so much invested in godels incompleteness
theorem
much maths is reliant on it but at the time godel wrote his theorem he
had
no idea of what truth was as peter smith the Cambridge expert on Godel
admitts

http://groups.google.com/group/sci.logic/browse_thread/thread/ebde70b...
de566912ee69f0a8?lnk=gst&q=G%C3%B6del+didn%27t+rely+on+the+notion+PETER
+smith#de 566912ee69f0a8

Quote:
[b]Gdel didn't rely on the notion
of truth[/b]

but truth is central to his theorem
as peter smith kindly tellls us

http://assets.cambridge.org/97805218...40_excerpt.pdf
Quote:
Godel did is find a general method that enabled him to take any theory
T strong enough to capture a modest amount of basic arithmetic and
construct a corresponding arithmetical sentence GT which encodes the
claim The sentence GT itself is unprovable in theory T. So G T is
[b]true[/b] if and only if T cant prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
[b]true[/b]-but-unprovable,
[/quote]

thus godels incompleteness theorem is about true statements which cant
be proven
but godel cant tell us what makes a mathematical statement true

thus his theorem is meaningless

Abram Demski
notes

http://omgili.com/newsgroups/sci/logic/5f72458d70b444f3fa127507e37747...

With no working definition of truth, Godels proof cannot be taken
through the last step which converts the formal result about sentences
that can and can't be proven into one about truth and
incompleteness.The formal result still holds, though; it is just of
questionable interest. Right?

in fact his theorem is meaningless as he cant tell us what truth is- Hide quoted text -

- Show quoted text -[/quote]

Daryl McCullough...

Posted: Thu Nov 05, 2009 11:31 am

Guest

Jesse F. Hughes says...
[quote]
Newberry <newberryxy at (no spam) gmail.com> writes:

Thirdly, you cannot have logic without the notion of truth.

It all depends on what you mean by logic, but as far as I can tell,
the syntax for FOL long preceded the semantics. Sure seems like they
had a logic back then.
[/quote]
I would say that, before semantics was developed in the sense of
model theory, there was an informal notion of truth. The rules of logic
were (presumably) developed under the constraint that they be
truth-preserving (if all the premises of a rule of inference are
true, then the conclusion must be true).

Of course, once you have a logic, you can treat it as a purely
syntactic game with symbols, and forget about notions of truth.
Or, presumably, you could just view it as a complicated production
system, a rule for generating strings in some language.

--
Daryl McCullough
Ithaca, NY

Rupert...

Posted: Thu Nov 05, 2009 6:23 pm

Guest

On Nov 5, 5:14 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
Rupert wrote:
On Nov 2, 6:24 pm, byron <spermato... at (no spam) yahoo.com> wrote:
the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
Gödel did in fact formulate a definition of truth independently of
Tarski, but this does not matter any way, because in fact he took to
formulate his theorem so that it only used syntactical notions. He
does not need the notion of truth.
I don't know what the Australian philosopher really said so I'm not
going to defend him. But to say that Godel did "not need the notion
of truth" in his Incompleteness work is really wrong! Without the
(_perceived_) notion of arithmetic truths Godel's work would tantamount
to nothing.
No, it's not true that Godel "_only_ used syntactical notions" in his
work. Far from it!
I wrote "he took to formulate his theorem so that it only used
syntactical notions". And of course I meant to write "he took care to
formulate his theorem so that it only used syntactical notions." This
is quite indisputable. Have you read the paper?
Yes, I read a translated paper. Have you _carefully_ read any?
Yes.
In the translated version by B. Meltzer in the very first section Godel
had "... This situation is not due in some way to the special nature of
the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition
of a finite number of axioms to the two systems mentioned, provided
that thereby no false propositions of the kind described in footnote
4 become provable".
Note the phrase "..._false_ propositions...". If "false" has nothing
to do with "truth notion" then what is the notion of "false" about?
But in the penultimate paragraph of the first section he says "The
exact statement of the above proof, which now follows, will have among
others the task of substituting for the second of these assumptions a
purely formal and much weaker one." And then of course when you go on
to read the main part of the paper...
What he proves is that every omega-consistent primitive recursively
axiomatisable theory, whose language is an extension of the first-
order language of arithmetic, in which all the primitive recursive
functions are strongly representable, is incomplete. Omega-consistency
is the "purely formal and much weaker" assumption. It is a syntactic
condition, which allows him to dispense with the use of any semantic
notions. Later Rosser showed how to weaken the hypothesis of omega-
consistency to consistency.
the fact that the mathematical theorem
itself as formulated in the paper makes use of purely syntactic
notions - which is what I said - is beyond dispute.
His theorem is a _meta_ theorem, just in case you're not aware.
Indeed I was.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
(Otherwise they'd be just called
normal FOL theorems, derived from axioms.
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
That formulation is frequently used, yes, because it is somewhat
easier to explain, and furthermore he does use that formulation in the
initial informal exposition in the first part of the first section, as
you pointed out. But he goes on to emphasise that one could dispense
with the use of semantic notions, as I have just showed you above, and
this of course was precisely my point.
You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
Not at all.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
Yes it could. As mentioned above the conclusion of his theorem would
have "... _true_ but not provable". Without a notion of truth his
conclusion would be meaningless, having "true" appearing in the
conclusion.
No, not at all. If you have a copy of Meltzer's translation then
surely you have had a look at the actual formal statement of the
theorem? Go and have a look at the statement of Proposition VI.
His paper there (including Proposition VI) is _littered_ with "omega-
consistent", "recursive", natural numbers, arithmetical (Proposition VII),
and the like of _truths of the naturals_.
Omega-consistency is a syntactical notion.
Really?

Let me define the usual syntactical consistency or inconsistency
for, say, ZFC:

- ZFC is said to be syntactically inconsistent iff there exists
a formula written in L(ZF) of the form (F /\ ~F) which is
syntactically provable in ZFC.

- ZFC is said to be syntactically consistent iff it's not syntactically
inconsistent.

See. No notions such as "natural numbers", "arithmetic truth", etc...

Now, it's your turn to _syntactically_ define omega-consistency
or omega-inconsistency for ZFC - without any notions such as
"natural numbers", "arithmetic truth", etc...

Well, you have Meltzer's translation of the paper in front of you,
don't you? Why don't you just look up the definition?

In the case of ZFC, there is a natural scheme for translating
sentences in the first-order language of arithmetic into the first-
order language of set theory, right?

We say that ZFC is omega-inconsistent if for some arithmetical
predicate P(n) it proves EnP(n), but it also proves ~P(0), ~P(1), ~P
(2), and so on. That's the definition of omega-inconsistency. And ZFC
is omega-consistent if it is not omega-inconsistent.

But that's where you're wrong! You said before (right above):

>>> Omega-consistency is a syntactical notion.

Tell us then how ~P(0), ~P(1), ~P(2), _and so on_ could be syntactical?
What exactly did you mean for "so on" to be mathematically _syntactical_?

Do you really think the notion of natural numbers is a syntactical notion?- Hide quoted text -

- Show quoted text -
[/quote]
The procedure is: you substitute various numerals into the predicate P
(x). Do you think that the idea of a "numeral" is not a syntactical
notion?

Rupert...

Posted: Thu Nov 05, 2009 6:25 pm

Guest

On Nov 5, 4:00 pm, Newberry <newberr... at (no spam) gmail.com> wrote:
[quote]It is the notion that certain formulae are
true but not provable that makes the theorem interesting.

You are welcome to believe that if you want. I did not comment on
which version of the theorem is *interesting*; that is a subjective
matter. I observed that there *is* a syntactical version.

First of all the only reason we prove theorems is that they are
interesting.

Secondly, Goedel's sentence is either provable or unprovable. In the
first case it is true that it is provable and false that it is
unprovable. If it is unprovable it is very hard to avoid the
conclusion that Goedel's sentence is true. It is hard to leave the
truth out.

[/quote]
Not at all. *If* you include some notion of truth then all that will
follow. But the treatment can be purely syntactical, and Gödel takes
pains to show this.

[quote]Thirdly, you cannot have logic without the notion of truth.
[/quote]
Yes, you can.

Rupert...

Posted: Thu Nov 05, 2009 6:26 pm

Guest

On Nov 5, 5:32 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
I observed that there *is* a syntactical version.

Can you tell us all again who invented the truth version and who
the syntactical version?
[/quote]
Gödel discusses both versions in the introduction to his 1931 paper.

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