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| byron... |
 Posted: Sun Nov 01, 2009 9:24 pm |
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the australian philosopher colin leslie dean has shown Godel
incompleteness theorem meaningless as he has no idea
what truth is
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/ebde70bc932fc0a7/
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/5f72458d70b444f3fa127507e3774779localhosttalkaboutsciencecom.html
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 |
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| Rupert... |
| Posted: Mon Nov 02, 2009 8:19 am |
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Guest
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On Nov 2, 6:24 pm, 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]
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. |
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| Rupert... |
| Posted: Mon Nov 02, 2009 8:58 am |
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On Nov 3, 5:29 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]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!
[/quote]
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?
It could be defended that some philosophical applications of his work
make essential use of the notion of truth, and there is certainly no
doubt that Gödel himself was a realist and that this was a strong
motivation in all his work. But 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. You should have a
look at the paper if you doubt me.
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth. The result is a mathematical theorem
about a certain class of formal systems which are to be thought of as
idealised abstractions of the human practice of doing mathematics. In
this way the result gives us information about the limitations of
human mathematical practice, which is of interest.
If our esteemed Australian philosopher Colin Leslie Dean wishes to
reject the notion of mathematical truth then perhaps he should join
forces with Edward Nelson. |
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| Nam Nguyen... |
| Posted: Mon Nov 02, 2009 1:29 pm |
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Guest
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Rupert wrote:
[quote]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.
[/quote]
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! |
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| Rupert... |
| Posted: Mon Nov 02, 2009 5:50 pm |
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Guest
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On Nov 3, 10:22 am, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]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?
[/quote]
Yes.
[quote]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?
[/quote]
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.
[quote]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.
[/quote]
Indeed I was.
[quote]Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation.
[/quote]
Why would that be? What about the deduction theorem, or the cut-
elimination theorem?
[quote](Otherwise they'd be just called
normal FOL theorems, derived from axioms.
[/quote]
They *are* derived from axioms. Metatheorems are proved in a
metatheory.
[quote]Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
[/quote]
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.
[quote]You should have a look at the paper if you doubt me.
Actually, you should have really doubt what you've claimed.
[/quote]
Not at all.
[quote]
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.
[/quote]
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.
[quote]
- Show quoted text -[/quote] |
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| Nam Nguyen... |
| Posted: Mon Nov 02, 2009 6:22 pm |
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Guest
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Rupert wrote:
[quote]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?
[/quote]
Yes, I read a translated paper. Have you _carefully_ read any?
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?
[quote]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.
[/quote]
His theorem is a _meta_ theorem, just in case you're not aware.
Any any meta theorem about FOL systems _will be_ more than just
syntactical in its formulation. (Otherwise they'd be just called
normal FOL theorems, derived from axioms. Isn't it true that as
part of his formulation, one would typically see "... _true_
but not provable"?)
[quote]You should have a look at the paper if you doubt me.
[/quote]
Actually, you should have really doubt what you've claimed.
[quote]
And it could not be defended that his work amounts to nothing in the
absence of a notion of truth.
[/quote]
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. |
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| Newberry... |
| Posted: Mon Nov 02, 2009 8:39 pm |
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Guest
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On Nov 2, 7:50 pm, Rupert <rupertmccal... at (no spam) yahoo.com> wrote:
[quote]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.
[/quote]
I thik that you are wrong. It is the notion that certain formulae are
true but not provable that makes the theorem interesting. It is
certainly this notion that generates all the hoopla.
Secondly according to classical logic A v ~A. So if the system is
incomplete then some true sentences are necessarily unprovable.
[quote]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.
- Show quoted text -- Hide quoted text -
- Show quoted text -- Hide quoted text -
- Show quoted text -[/quote] |
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| Nam Nguyen... |
| Posted: Mon Nov 02, 2009 11:19 pm |
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Guest
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Rupert wrote:
[quote]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.
[/quote]
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_.
Why don't you rewrite his (finite) paper by deleting out all of those
words and see how far you could "prove" his Incompleteness Theorem(s)?
If you can't, then you just got mixed up between what's syntactical
and what's truth.
Let me re-emphasize this fact to you: once you must depend on the naturals
in one form or another (recursion, omega-consistent, etc...), your proof
will NOT be syntactical! Syntactical proof stands on its own and never
depends on natural numbers. But you cant' prove consistency of a formal
system using purely syntactical means: rules of inference is for proving
not for disproving (which you'd need to prove in meta level any kind of
syntactical consistency).
So again, why don't you re-trace Godel's work purely syntactically?
(I.e without mentioning about the naturals or the like of it at all). |
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| James R Meyer... |
| Posted: Tue Nov 03, 2009 2:58 am |
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Guest
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It is actually quite simple to show that Colin Dean is wide of the
mark. He needs to appreciate the difference between finding a
fundamental flaw in the logic of a proof and quibbling over aspects of
it that are not absolutely essential to the proof. I don't know why
people who want to disagree with him don't simply point out his
errors. You can see the reasons why Dean's argument is wrong at:
http://jamesrmeyer.com/ffgit/show_error.html |
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| Rupert... |
| Posted: Tue Nov 03, 2009 9:11 am |
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Guest
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On Nov 3, 3:19 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]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_.
[/quote]
Omega-consistency is a syntactical notion.
[quote]Why don't you rewrite his (finite) paper by deleting out all of those
words and see how far you could "prove" his Incompleteness Theorem(s)?
If you can't, then you just got mixed up between what's syntactical
and what's truth.
Let me re-emphasize this fact to you: once you must depend on the naturals
in one form or another (recursion, omega-consistent, etc...), your proof
will NOT be syntactical! Syntactical proof stands on its own and never
depends on natural numbers. But you cant' prove consistency of a formal
system using purely syntactical means: rules of inference is for proving
not for disproving (which you'd need to prove in meta level any kind of
syntactical consistency).
So again, why don't you re-trace Godel's work purely syntactically?
(I.e without mentioning about the naturals or the like of it at all).- Hide quoted text -
- Show quoted text -[/quote] |
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| Rupert... |
| Posted: Tue Nov 03, 2009 9:14 am |
|
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Guest
|
On Nov 3, 5:39 pm, Newberry <newberr... at (no spam) gmail.com> wrote:
[quote]On Nov 2, 7:50 pm, Rupert <rupertmccal... at (no spam) yahoo.com> 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.
I thik that you are wrong.
[/quote]
Which bit of the above is wrong?
[quote]It is the notion that certain formulae are
true but not provable that makes the theorem interesting.
[/quote]
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]It is
certainly this notion that generates all the hoopla.
Secondly according to classical logic A v ~A. So if the system is
incomplete then some true sentences are necessarily unprovable.
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.
- Show quoted text -- Hide quoted text -
- Show quoted text -- Hide quoted text -
- Show quoted text -- Hide quoted text -
- Show quoted text -- Hide quoted text -
- Show quoted text -[/quote] |
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| Rupert... |
| Posted: Tue Nov 03, 2009 9:30 pm |
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Guest
|
On Nov 4, 4:49 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]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...
[/quote]
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. |
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| Rupert... |
| Posted: Tue Nov 03, 2009 9:31 pm |
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Guest
|
On Nov 4, 5:00 pm, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote]Rupert wrote:
On Nov 3, 5:39 pm, Newberry <newberr... at (no spam) gmail.com> wrote:
On Nov 2, 7:50 pm, Rupert <rupertmccal... at (no spam) yahoo.com> 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.
I thik that you are wrong.
Which bit of the above is wrong?
My guess would be this bit about what he (Godel) did:
>>>>>>> in fact he took to formulate his theorem so that it only used
>>>>>>> syntactical notions. He does not need the notion of truth.
Godel's formulation of his meta theorems requires notion of truth of
the natural numbers, in one way or another.
[/quote]
No.
You've *said* this plenty of times, but a reading of the paper which
you claim to have read will reveal that you're wrong. |
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| Nam Nguyen... |
| Posted: Wed Nov 04, 2009 12:49 am |
|
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Guest
|
Rupert wrote:
[quote]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.
[/quote]
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...
[quote]
Why don't you rewrite his (finite) paper by deleting out all of those
words and see how far you could "prove" his Incompleteness Theorem(s)?
If you can't, then you just got mixed up between what's syntactical
and what's truth.
Let me re-emphasize this fact to you: once you must depend on the naturals
in one form or another (recursion, omega-consistent, etc...), your proof
will NOT be syntactical! Syntactical proof stands on its own and never
depends on natural numbers. But you cant' prove consistency of a formal
system using purely syntactical means: rules of inference is for proving
not for disproving (which you'd need to prove in meta level any kind of
syntactical consistency).
So again, why don't you re-trace Godel's work purely syntactically?
(I.e without mentioning about the naturals or the like of it at all).- Hide quoted text -
- Show quoted text -
[/quote] |
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| Nam Nguyen... |
| Posted: Wed Nov 04, 2009 1:00 am |
|
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|
Guest
|
Rupert wrote:
[quote]On Nov 3, 5:39 pm, Newberry <newberr... at (no spam) gmail.com> wrote:
On Nov 2, 7:50 pm, Rupert <rupertmccal... at (no spam) yahoo.com> 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.
I thik that you are wrong.
Which bit of the above is wrong?
[/quote]
My guess would be this bit about what he (Godel) did:
[quote]in fact he took to formulate his theorem so that it only used
syntactical notions. He does not need the notion of truth.
[/quote]
Godel's formulation of his meta theorems requires notion of truth of
the natural numbers, in one way or another.
[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.
It is
certainly this notion that generates all the hoopla.
Secondly according to classical logic A v ~A. So if the system is
incomplete then some true sentences are necessarily unprovable.
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.
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