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From: Barb Knox on 20 Jun 2010 21:30
Since no-one has replied to you for 4 days I figured I'd have a go.

[added sci.logic]

(Note that opposition to Goedel's incompleteness theorems is a perennial
topic in sci.logic.)

In article
<9ee477fa-09ad-48bf-b8ea-1ac0bca640c1(a)42g2000prb.googlegroups.com>,
byron <spermatozon(a)yahoo.com> wrote:

> it is argued by colin leslie dean that no matter how faultless godels
> logic is Godels incompleteness theorem are invalid ie illegitimate
> for 5 reasons: he uses the axiom of reducibility- which is invalid ie
> illegitimate,he constructs impredicative statement which is invalid ie
> illegitimate ,he cant tell us what makes a mathematic statement true,
> he falls into two self-contradictions,he ends in three paradoxes
>
> <http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-invalid-illegitimate>
>
> http://gamahucherpress.yellowgum.com/gamahucher_press_catalogue.htm
>
> http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
>
> First of the two self-contradictions
>
> Godels first theorem ends in paradox �due to his construction of
> impredicative statement
> Now the syntactic version of Goedels first completeness theorem
> reads
>
> Proposition VI: To every ��-consistent recursive class c of
> formulae there correspond recursive class-signs r, such that neither v
> Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free
> variable of r).
>
> But when this is put into plain words we get
> http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem
>
> Goedel'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 truths, there is an arithmetical statement
> that is true,[1] but not provable in the theory (Kleene 1967, p. 250).
>
> Now truth in mathematics was considered to be if a statement can
> be proven then it is true
> Ie truth is equated with provability

Sorry, but that's just flat wrong.

> http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
> ��from at least the time of Hilbert's program at the turn of the
> twentieth century to the proof of Goedel's theorem and the development
> of the Church-Turing thesis in the early part of that century, true
> statements in mathematics were generally assumed to be those
> statements which are provable in a formal axiomatic system.
> The works of Kurt Goedel, Alan Turing, and others shook this
> assumption, with the development of statements that are true but
> cannot be proven within the system�

So, truth WAS historically equated with provability, until Goedel at al
showed that these were in fact distinct. You can hardly use this
historical fact as an argument against Goedel at al.

The next section after your linked one is "Semantic theory of truth".
This is the current common mathematical usage of "Truth". You would
benefit from reading and understanding it.


> http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem
> �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 truths, there is an arithmetical statement
> that is true,[1] but not provable in the theory (Kleene 1967, p. 250)
> For each consistent formal theory T having the required small
> amount of number theory
> � provability-within-the-theory-T is not the same as truth; the
> theory T is incomplete.�
>
> Now it is said godel PROVED
> "there are true mathematical statements which cant be proven"
> in other words
> truth does not equate with proof.
>
> if that theorem is true
> then his theorem is false
>
> PROOF
> for if the theorem is true
> then truth does equate with proof-
> as he has given proof of a true statement

Oops. You've gone totally off the rails here. Goedel gives *a* proof
of something, just as all mathematicians do. In the proof uses the
assumption that the formal system is consistent, i.e. that if something
is provable in that system then it is true (in this case in the Standard
Model of arithmetic). So Goedel uses the *assumption* that proof
implies truth in his (meta-)proof; he most emphatically does NOT assume
that truth implies proof, indeed the whole point of the proof is that it
does not.


> but his theorem says
> truth does not equate with proof.
> thus a paradox
> THIS WHAT COMES OF USING IMPREDICATIVE STATEMENTS
>
>
> GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
>
> GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
> Now truth in mathematics was considered to be if a statement can
> be proven then it is true
> Ie truth was s equated with provability
> http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics

Yes, "WAS considered". No longer.

[redundant quotes snipped]

> For each consistent formal theory T having the required small
> amount of number theory
> �c provability-within-the-theory-T is not the same as truth; the
> theory T is incomplete.
>
> In other words there are true mathematical statements which cant
> be proven
> But the fact is Goedel cant tell us what makes a mathematical
> statement true thus his theorem is meaningless

That is again just flat wrong. Tarski's work on semantical truth was
current at the time of Goedel's work. Goedel intentionally wanted a
purely syntactic demonstration, not one that relied on semantic truth.
Goedel knew perfectly well what makes a mathematical statement
semantically true in the modern sense. His incompleteness theorems are
simply not about that.

> Ie if Godels theorem said there were gibbly statements that cant
> be proven
> But if Goedel cant tell us what a gibbly statement was then we
> would say his theorem was meaningless

In his proof he *constructs* (or rather gives the procedure for
constructing) a "gibbly" statement. That is the core of the proof! Not
only does he tell us what it "is", he actually shows how to build one.


> Now at the time godel wrote his theorem he had no idea of what
> truth was as peter smith the Cambridge expert on Godel admitts

Utter rubbish. Goedel earlier work was with model theory (i.e. semantic
truth); he got his PhD for the "completeness theorem" for first-order
logic.

> http://groups.google.com/group/sci.logi ... 12ee69f0a8
>
> Quote:
> Goedel didn't rely on the notion of truth

Yes, he didn't *rely* on it in his incompleteness theorems,
intentionally. That certainly does not imply he was ignorant of it.


> 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 true if and only
> if T can�t 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
> true-but-unprovable,
>
> and godels theorem is
>
> http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir...
> Quote:
> Goedel's first incompleteness theorem, perhaps the single most
> celebrated result in mathematical logic, states that:
>
> For any consistent formal, recursively enumerable theory that
> proves basic arithmetical truths, an arithmetical statement that is
> true, but not provable in the theory, can be constructed.1 That is,
> any effectively generated theory capable of expressing elementary
> arithmetic cannot be both consistent and complete.
>
> you see godel referes to true statement

Even though popularisations of Goedel's results discuss them in terms of
"truth", the results themselves do not. The results are about
*incompleteness* -- there is a proposition G such that neither G nor
not-G can be proven. This has implications involving mathematical truth
(e.g. that there are non-standard models of arithmetic), but the
theorems themselves are not about mathematical truth.


> but Goedel didn't rely on the notion
> of truth
>
> now because Goedel didn't rely on the notion
> of truth he cant tell us what true statements are
> thus his theorem is meaningless


--
---------------------------
| 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 videtur.
| BBB aa a r bbb |
-----------------------------
From: Newberry on 21 Jun 2010 00:10
On Jun 20, 6:30 pm, Barb Knox <s...(a)sig.below> wrote:
> Since no-one has replied to you for 4 days I figured I'd have a go.
>
> [added sci.logic]
>
> (Note that opposition to Goedel's incompleteness theorems is a perennial
> topic in sci.logic.)
>
> In article
> <9ee477fa-09ad-48bf-b8ea-1ac0bca64...(a)42g2000prb.googlegroups.com>,
>
>
>
>
>
>  byron <spermato...(a)yahoo.com> wrote:
> > it is argued by colin leslie dean that no matter how faultless godels
> > logic is  Godels incompleteness theorem are invalid ie illegitimate
> > for 5 reasons: he uses the axiom of reducibility- which is invalid ie
> > illegitimate,he constructs impredicative statement which is invalid ie
> > illegitimate ,he cant tell us what makes a mathematic statement true,
> > he falls into two self-contradictions,he ends in three paradoxes
>
> > <http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-inva....>
>
> >http://gamahucherpress.yellowgum.com/gamahucher_press_catalogue.htm
>
> >http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
>
> > First of the two self-contradictions
>
> >     Godels first theorem ends in paradox due to his construction of
> > impredicative statement
> >     Now the syntactic version of Goedels first completeness theorem
> > reads
>
> >     Proposition VI: To every -consistent recursive class c of
> > formulae there correspond recursive class-signs r, such that neither v
> > Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free
> > variable of r).
>
> >     But when this is put into plain words we get
> >    http://en.wikipedia.org/wiki/G%C3%B6del... ss_theorem
>
> >     Goedel'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 truths, there is an arithmetical statement
> > that is true,[1] but not provable in the theory (Kleene 1967, p. 250).
>
> >     Now truth in mathematics was considered to be if a statement can
> > be proven then it is true
> >     Ie truth is equated with provability
>
> Sorry, but that's just flat wrong.
>
> >    http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
> >     from at least the time of Hilbert's program at the turn of the
> > twentieth century to the proof of Goedel's theorem and the development
> > of the Church-Turing thesis in the early part of that century, true
> > statements in mathematics were generally assumed to be those
> > statements which are provable in a formal axiomatic system.
> >     The works of Kurt Goedel, Alan Turing, and others shook this
> > assumption, with the development of statements that are true but
> > cannot be proven within the system
>
> So, truth WAS historically equated with provability, until Goedel at al
> showed that these were in fact distinct.  You can hardly use this
> historical fact as an argument against Goedel at al.
>
> The next section after your linked one is "Semantic theory of truth".  
> This is the current common mathematical usage of "Truth".  You would
> benefit from reading and understanding it.
>
>
>
>
>
> >    http://en.wikipedia.org/wiki/G%C3%B6del... ss_theorem
> >     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 truths, there is an arithmetical statement
> > that is true,[1] but not provable in the theory (Kleene 1967, p. 250)
> >     For each consistent formal theory T having the required small
> > amount of number theory
> >     provability-within-the-theory-T is not the same as truth; the
> > theory T is incomplete.
>
> >     Now it is said godel PROVED
> >     "there are true mathematical statements which cant be proven"
> >     in other words
> >     truth does not equate with proof.
>
> >     if that theorem is true
> >     then his theorem is false
>
> >     PROOF
> >     for if the theorem is true
> >     then truth does equate with proof-
> >     as he has given proof of a true statement
>
> Oops.  You've gone totally off the rails here.  Goedel gives *a* proof
> of something, just as all mathematicians do.  In the proof uses the
> assumption that the formal system is consistent, i.e. that if something
> is provable in that system then it is true (in this case in the Standard
> Model of arithmetic).  So Goedel uses the *assumption* that proof
> implies truth in his (meta-)proof; he most emphatically does NOT assume
> that truth implies proof, indeed the whole point of the proof is that it
> does not.
>
> >     but his theorem says
> >     truth does not equate with proof.
> >     thus a paradox
> >     THIS WHAT COMES OF USING IMPREDICATIVE STATEMENTS
>
> > GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
>
> >     GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
> >     Now truth in mathematics was considered to be if a statement can
> > be proven then it is true
> >     Ie truth was s equated with provability
> >    http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
>
> Yes, "WAS considered".  No longer.
>
> [redundant quotes snipped]
>
> >     For each consistent formal theory T having the required small
> > amount of number theory
> >     c provability-within-the-theory-T is not the same as truth; the
> > theory T is incomplete.
>
> >     In other words there are true mathematical statements which cant
> > be proven
> >     But the fact is Goedel cant tell us what makes a mathematical
> > statement true thus his theorem is meaningless
>
> That is again just flat wrong.  Tarski's work on semantical truth was
> current at the time of Goedel's work.  Goedel intentionally wanted a
> purely syntactic demonstration, not one that relied on semantic truth.  
> Goedel knew perfectly well what makes a mathematical statement
> semantically true in the modern sense.  His incompleteness theorems are
> simply not about that.
>
> >     Ie if Godels theorem said there were gibbly statements that cant
> > be proven
> >     But if Goedel cant tell us what a gibbly statement was then we
> > would say his theorem was meaningless
>
> In his proof he *constructs* (or rather gives the procedure for
> constructing) a "gibbly" statement.  That is the core of the proof!  Not
> only does he tell us what it "is", he actually shows how to build one.
>
> >     Now at the time godel wrote his theorem he had no idea of what
> > truth was as peter smith the Cambridge expert on Godel admitts
>
> Utter rubbish.  Goedel earlier work was with model theory (i.e. semantic
> truth); he got his PhD for the "completeness theorem" for first-order
> logic.
>
> >    http://groups.google.com/group/sci.logi... 12ee69f0a8
>
> >     Quote:
> >     Goedel didn't rely on the notion of truth
>
> Yes, he didn't *rely* on it in his incompleteness theorems,
> intentionally.  That certainly does not imply he was ignorant of it.
>
>
>
>
>
> >     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 true if and only
> >     if T can t 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
> >     true-but-unprovable,
>
> >     and godels theorem is
>
> >    http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir...
> >     Quote:
> >     Goedel's first incompleteness theorem, perhaps the single most
> > celebrated result in mathematical logic, states that:
>
> >     For any consistent formal, recursively enumerable theory that
> > proves basic arithmetical truths, an arithmetical statement that is
> > true, but not provable in the theory, can be constructed.1 That is,
> > any effectively generated theory capable of expressing elementary
> > arithmetic cannot be both consistent and complete.
>
> >     you see godel referes to true statement
>
> Even though popularisations of Goedel's results discuss them in terms of
> "truth", the results themselves do not.  The results are about
> *incompleteness* -- there is a proposition G such that neither G nor
> not-G can be proven.  This has implications involving mathematical truth
> (e.g. that there are non-standard models of arithmetic), but the
> theorems themselves are not about mathematical truth.

We better get this straight. Thie is from Wikipedia:

"for any consistent, effectively generated formal theory that proves
certain basic arithmetic truths, there is an arithmetical statement
that is true,[1] but not provable in the theory" (Kleene 1967, p.
250).

http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Is this a popularization? Many textbooks state the theorem in terms of
truth and provability.

I do not know how Kleene would prove that in ANY conceivable theory
capable of arithmetic with a plausible interpretation there are
unprovable truths. (Although I tend to believe that "there is a
proposition G such that neither G nor not-G can be proven" holds for
ANY theory capable of arithmetic.)

> >     but Goedel didn't rely on the notion
> >     of truth
>
> >     now because Goedel didn't rely on the notion
> >     of truth he cant tell us what true statements are
> >     thus his theorem is meaningless
>
> --
> ---------------------------
> |  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 videtur.
> |  BBB    aa a  r     bbb   |  
> ------------------------------ Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

From: Rupert on 21 Jun 2010 01:58
On Jun 21, 2:10 pm, Newberry <newberr...(a)gmail.com> wrote:
> On Jun 20, 6:30 pm, Barb Knox <s...(a)sig.below> wrote:
>
> > Since no-one has replied to you for 4 days I figured I'd have a go.
>
> > [added sci.logic]
>
> > (Note that opposition to Goedel's incompleteness theorems is a perennial
> > topic in sci.logic.)
>
> > In article
> > <9ee477fa-09ad-48bf-b8ea-1ac0bca64...(a)42g2000prb.googlegroups.com>,
>
> >  byron <spermato...(a)yahoo.com> wrote:
> > > it is argued by colin leslie dean that no matter how faultless godels
> > > logic is  Godels incompleteness theorem are invalid ie illegitimate
> > > for 5 reasons: he uses the axiom of reducibility- which is invalid ie
> > > illegitimate,he constructs impredicative statement which is invalid ie
> > > illegitimate ,he cant tell us what makes a mathematic statement true,
> > > he falls into two self-contradictions,he ends in three paradoxes
>
> > > <http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-inva...>
>
> > >http://gamahucherpress.yellowgum.com/gamahucher_press_catalogue.htm
>
> > >http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
>
> > > First of the two self-contradictions
>
> > >     Godels first theorem ends in paradox due to his construction of
> > > impredicative statement
> > >     Now the syntactic version of Goedels first completeness theorem
> > > reads
>
> > >     Proposition VI: To every -consistent recursive class c of
> > > formulae there correspond recursive class-signs r, such that neither v
> > > Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free
> > > variable of r).
>
> > >     But when this is put into plain words we get
> > >    http://en.wikipedia.org/wiki/G%C3%B6del... ss_theorem
>
> > >     Goedel'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 truths, there is an arithmetical statement
> > > that is true,[1] but not provable in the theory (Kleene 1967, p. 250)..
>
> > >     Now truth in mathematics was considered to be if a statement can
> > > be proven then it is true
> > >     Ie truth is equated with provability
>
> > Sorry, but that's just flat wrong.
>
> > >    http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
> > >     from at least the time of Hilbert's program at the turn of the
> > > twentieth century to the proof of Goedel's theorem and the development
> > > of the Church-Turing thesis in the early part of that century, true
> > > statements in mathematics were generally assumed to be those
> > > statements which are provable in a formal axiomatic system.
> > >     The works of Kurt Goedel, Alan Turing, and others shook this
> > > assumption, with the development of statements that are true but
> > > cannot be proven within the system
>
> > So, truth WAS historically equated with provability, until Goedel at al
> > showed that these were in fact distinct.  You can hardly use this
> > historical fact as an argument against Goedel at al.
>
> > The next section after your linked one is "Semantic theory of truth".  
> > This is the current common mathematical usage of "Truth".  You would
> > benefit from reading and understanding it.
>
> > >    http://en.wikipedia.org/wiki/G%C3%B6del... ss_theorem
> > >     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 truths, there is an arithmetical statement
> > > that is true,[1] but not provable in the theory (Kleene 1967, p. 250)
> > >     For each consistent formal theory T having the required small
> > > amount of number theory
> > >     provability-within-the-theory-T is not the same as truth; the
> > > theory T is incomplete.
>
> > >     Now it is said godel PROVED
> > >     "there are true mathematical statements which cant be proven"
> > >     in other words
> > >     truth does not equate with proof.
>
> > >     if that theorem is true
> > >     then his theorem is false
>
> > >     PROOF
> > >     for if the theorem is true
> > >     then truth does equate with proof-
> > >     as he has given proof of a true statement
>
> > Oops.  You've gone totally off the rails here.  Goedel gives *a* proof
> > of something, just as all mathematicians do.  In the proof uses the
> > assumption that the formal system is consistent, i.e. that if something
> > is provable in that system then it is true (in this case in the Standard
> > Model of arithmetic).  So Goedel uses the *assumption* that proof
> > implies truth in his (meta-)proof; he most emphatically does NOT assume
> > that truth implies proof, indeed the whole point of the proof is that it
> > does not.
>
> > >     but his theorem says
> > >     truth does not equate with proof.
> > >     thus a paradox
> > >     THIS WHAT COMES OF USING IMPREDICATIVE STATEMENTS
>
> > > GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
>
> > >     GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
> > >     Now truth in mathematics was considered to be if a statement can
> > > be proven then it is true
> > >     Ie truth was s equated with provability
> > >    http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
>
> > Yes, "WAS considered".  No longer.
>
> > [redundant quotes snipped]
>
> > >     For each consistent formal theory T having the required small
> > > amount of number theory
> > >     c provability-within-the-theory-T is not the same as truth; the
> > > theory T is incomplete.
>
> > >     In other words there are true mathematical statements which cant
> > > be proven
> > >     But the fact is Goedel cant tell us what makes a mathematical
> > > statement true thus his theorem is meaningless
>
> > That is again just flat wrong.  Tarski's work on semantical truth was
> > current at the time of Goedel's work.  Goedel intentionally wanted a
> > purely syntactic demonstration, not one that relied on semantic truth.  
> > Goedel knew perfectly well what makes a mathematical statement
> > semantically true in the modern sense.  His incompleteness theorems are
> > simply not about that.
>
> > >     Ie if Godels theorem said there were gibbly statements that cant
> > > be proven
> > >     But if Goedel cant tell us what a gibbly statement was then we
> > > would say his theorem was meaningless
>
> > In his proof he *constructs* (or rather gives the procedure for
> > constructing) a "gibbly" statement.  That is the core of the proof!  Not
> > only does he tell us what it "is", he actually shows how to build one.
>
> > >     Now at the time godel wrote his theorem he had no idea of what
> > > truth was as peter smith the Cambridge expert on Godel admitts
>
> > Utter rubbish.  Goedel earlier work was with model theory (i.e. semantic
> > truth); he got his PhD for the "completeness theorem" for first-order
> > logic.
>
> > >    http://groups.google.com/group/sci.logi... 12ee69f0a8
>
> > >     Quote:
> > >     Goedel didn't rely on the notion of truth
>
> > Yes, he didn't *rely* on it in his incompleteness theorems,
> > intentionally.  That certainly does not imply he was ignorant of it.
>
> > >     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 true if and only
> > >     if T can t 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
> > >     true-but-unprovable,
>
> > >     and godels theorem is
>
> > >    http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir...
> > >     Quote:
> > >     Goedel's first incompleteness theorem, perhaps the single most
> > > celebrated result in mathematical logic, states that:
>
> > >     For any consistent formal, recursively enumerable theory that
> > > proves basic arithmetical truths, an arithmetical statement that is
> > > true, but not provable in the theory, can be constructed.1 That is,
> > > any effectively generated theory capable of expressing elementary
> > > arithmetic cannot be both consistent and complete.
>
> > >     you see godel referes to true statement
>
> > Even though popularisations of Goedel's results discuss them in terms of
> > "truth", the results themselves do not.  The results are about
> > *incompleteness* -- there is a proposition G such that neither G nor
> > not-G can be proven.  This has implications involving mathematical truth
> > (e.g. that there are non-standard models of arithmetic), but the
> > theorems themselves are not about mathematical truth.
>
> We better get this straight. Thie is from Wikipedia:
>
> "for any consistent, effectively generated formal theory that proves
> certain basic arithmetic truths, there is an arithmetical statement
> that is true,[1] but not provable in the theory" (Kleene 1967, p.
> 250).
>
> http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
>
> Is this a popularization? Many textbooks state the theorem in terms of
> truth and provability.
>

There is the syntactic version and the semantic version.

The semantic version is "Given any primitive recursively axiomatisable
theory T in a language in which the first-order language of arithmetic
can be interpreted, such that T is arithmetically sound, there exist
true sentences in the first-order language of arithmetic not provable
in T." Goedel informally argues for this conclusion in the
introduction to the paper.

The syntactic version is "Given any primitive recursively
axiomatisable theory T in a language in which the first-order language
of arithmetic can be interpreted, such that all primitive recursive
functions can be represented in T and T is omega-consistent, there
exist sentences in the first-order langauge of arithmetic which are
independent of T." This is the version argued for in detail in the
main text of the paper.

> I do not know how Kleene would prove that in ANY conceivable theory
> capable of arithmetic with a plausible interpretation there are
> unprovable truths. (Although I tend to believe that "there is a
> proposition G such that neither G nor not-G can be proven" holds for
> ANY theory capable of arithmetic.)> >     but Goedel didn't rely on the notion
> > >     of truth
>
> > >     now because Goedel didn't rely on the notion
> > >     of truth he cant tell us what true statements are
> > >     thus his theorem is meaningless
>
> > --
> > ---------------------------
> > |  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 videtur.
> > |  BBB    aa a  r     bbb   |  
> > ------------------------------ Hide quoted text -
>
> > - Show quoted text -- Hide quoted text -
>
> > - Show quoted text -- Hide quoted text -
>
> > - Show quoted text -

From: herbzet on 21 Jun 2010 21:42


Rupert wrote:

> There is the syntactic version and the semantic version.
>
> The semantic version is "Given any primitive recursively axiomatisable
> theory T in a language in which the first-order language of arithmetic
> can be interpreted, such that T is arithmetically sound, there exist
> true sentences in the first-order language of arithmetic not provable
> in T." Goedel informally argues for this conclusion in the
> introduction to the paper.
>
> The syntactic version is "Given any primitive recursively
> axiomatisable theory T in a language in which the first-order language
> of arithmetic can be interpreted, such that all primitive recursive
> functions can be represented in T and T is omega-consistent, there
> exist sentences in the first-order langauge of arithmetic which are
> independent of T." This is the version argued for in detail in the
> main text of the paper.

Nice, concise -- thank you.

--
hz
From: Newberry on 21 Jun 2010 23:52
On Jun 20, 10:58 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Jun 21, 2:10 pm, Newberry <newberr...(a)gmail.com> wrote:
>
> > On Jun 20, 6:30 pm, Barb Knox <s...(a)sig.below> wrote:
>
> > > Since no-one has replied to you for 4 days I figured I'd have a go.
>
> > > [added sci.logic]
>
> > > (Note that opposition to Goedel's incompleteness theorems is a perennial
> > > topic in sci.logic.)
>
> > > In article
> > > <9ee477fa-09ad-48bf-b8ea-1ac0bca64...(a)42g2000prb.googlegroups.com>,
>
> > >  byron <spermato...(a)yahoo.com> wrote:
> > > > it is argued by colin leslie dean that no matter how faultless godels
> > > > logic is  Godels incompleteness theorem are invalid ie illegitimate
> > > > for 5 reasons: he uses the axiom of reducibility- which is invalid ie
> > > > illegitimate,he constructs impredicative statement which is invalid ie
> > > > illegitimate ,he cant tell us what makes a mathematic statement true,
> > > > he falls into two self-contradictions,he ends in three paradoxes
>
> > > > <http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-inva...>
>
> > > >http://gamahucherpress.yellowgum.com/gamahucher_press_catalogue.htm
>
> > > >http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
>
> > > > First of the two self-contradictions
>
> > > >     Godels first theorem ends in paradox due to his construction of
> > > > impredicative statement
> > > >     Now the syntactic version of Goedels first completeness theorem
> > > > reads
>
> > > >     Proposition VI: To every -consistent recursive class c of
> > > > formulae there correspond recursive class-signs r, such that neither v
> > > > Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free
> > > > variable of r).
>
> > > >     But when this is put into plain words we get
> > > >    http://en.wikipedia.org/wiki/G%C3%B6del... ss_theorem
>
> > > >     Goedel'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 truths, there is an arithmetical statement
> > > > that is true,[1] but not provable in the theory (Kleene 1967, p. 250).
>
> > > >     Now truth in mathematics was considered to be if a statement can
> > > > be proven then it is true
> > > >     Ie truth is equated with provability
>
> > > Sorry, but that's just flat wrong.
>
> > > >    http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
> > > >     from at least the time of Hilbert's program at the turn of the
> > > > twentieth century to the proof of Goedel's theorem and the development
> > > > of the Church-Turing thesis in the early part of that century, true
> > > > statements in mathematics were generally assumed to be those
> > > > statements which are provable in a formal axiomatic system.
> > > >     The works of Kurt Goedel, Alan Turing, and others shook this
> > > > assumption, with the development of statements that are true but
> > > > cannot be proven within the system
>
> > > So, truth WAS historically equated with provability, until Goedel at al
> > > showed that these were in fact distinct.  You can hardly use this
> > > historical fact as an argument against Goedel at al.
>
> > > The next section after your linked one is "Semantic theory of truth".  
> > > This is the current common mathematical usage of "Truth".  You would
> > > benefit from reading and understanding it.
>
> > > >    http://en.wikipedia.org/wiki/G%C3%B6del... ss_theorem
> > > >     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 truths, there is an arithmetical statement
> > > > that is true,[1] but not provable in the theory (Kleene 1967, p. 250)
> > > >     For each consistent formal theory T having the required small
> > > > amount of number theory
> > > >     provability-within-the-theory-T is not the same as truth; the
> > > > theory T is incomplete.
>
> > > >     Now it is said godel PROVED
> > > >     "there are true mathematical statements which cant be proven"
> > > >     in other words
> > > >     truth does not equate with proof.
>
> > > >     if that theorem is true
> > > >     then his theorem is false
>
> > > >     PROOF
> > > >     for if the theorem is true
> > > >     then truth does equate with proof-
> > > >     as he has given proof of a true statement
>
> > > Oops.  You've gone totally off the rails here.  Goedel gives *a* proof
> > > of something, just as all mathematicians do.  In the proof uses the
> > > assumption that the formal system is consistent, i.e. that if something
> > > is provable in that system then it is true (in this case in the Standard
> > > Model of arithmetic).  So Goedel uses the *assumption* that proof
> > > implies truth in his (meta-)proof; he most emphatically does NOT assume
> > > that truth implies proof, indeed the whole point of the proof is that it
> > > does not.
>
> > > >     but his theorem says
> > > >     truth does not equate with proof.
> > > >     thus a paradox
> > > >     THIS WHAT COMES OF USING IMPREDICATIVE STATEMENTS
>
> > > > GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
>
> > > >     GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
> > > >     Now truth in mathematics was considered to be if a statement can
> > > > be proven then it is true
> > > >     Ie truth was s equated with provability
> > > >    http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
>
> > > Yes, "WAS considered".  No longer.
>
> > > [redundant quotes snipped]
>
> > > >     For each consistent formal theory T having the required small
> > > > amount of number theory
> > > >     c provability-within-the-theory-T is not the same as truth; the
> > > > theory T is incomplete.
>
> > > >     In other words there are true mathematical statements which cant
> > > > be proven
> > > >     But the fact is Goedel cant tell us what makes a mathematical
> > > > statement true thus his theorem is meaningless
>
> > > That is again just flat wrong.  Tarski's work on semantical truth was
> > > current at the time of Goedel's work.  Goedel intentionally wanted a
> > > purely syntactic demonstration, not one that relied on semantic truth..  
> > > Goedel knew perfectly well what makes a mathematical statement
> > > semantically true in the modern sense.  His incompleteness theorems are
> > > simply not about that.
>
> > > >     Ie if Godels theorem said there were gibbly statements that cant
> > > > be proven
> > > >     But if Goedel cant tell us what a gibbly statement was then we
> > > > would say his theorem was meaningless
>
> > > In his proof he *constructs* (or rather gives the procedure for
> > > constructing) a "gibbly" statement.  That is the core of the proof!  Not
> > > only does he tell us what it "is", he actually shows how to build one..
>
> > > >     Now at the time godel wrote his theorem he had no idea of what
> > > > truth was as peter smith the Cambridge expert on Godel admitts
>
> > > Utter rubbish.  Goedel earlier work was with model theory (i.e. semantic
> > > truth); he got his PhD for the "completeness theorem" for first-order
> > > logic.
>
> > > >    http://groups.google.com/group/sci.logi... 12ee69f0a8
>
> > > >     Quote:
> > > >     Goedel didn't rely on the notion of truth
>
> > > Yes, he didn't *rely* on it in his incompleteness theorems,
> > > intentionally.  That certainly does not imply he was ignorant of it..
>
> > > >     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 true if and only
> > > >     if T can t 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
> > > >     true-but-unprovable,
>
> > > >     and godels theorem is
>
> > > >    http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir...
> > > >     Quote:
> > > >     Goedel's first incompleteness theorem, perhaps the single most
> > > > celebrated result in mathematical logic, states that:
>
> > > >     For any consistent formal, recursively enumerable theory that
> > > > proves basic arithmetical truths, an arithmetical statement that is
> > > > true, but not provable in the theory, can be constructed.1 That is,
> > > > any effectively generated theory capable of expressing elementary
> > > > arithmetic cannot be both consistent and complete.
>
> > > >     you see godel referes to true statement
>
> > > Even though popularisations of Goedel's results discuss them in terms of
> > > "truth", the results themselves do not.  The results are about
> > > *incompleteness* -- there is a proposition G such that neither G nor
> > > not-G can be proven.  This has implications involving mathematical truth
> > > (e.g. that there are non-standard models of arithmetic), but the
> > > theorems themselves are not about mathematical truth.
>
> > We better get this straight. Thie is from Wikipedia:
>
> > "for any consistent, effectively generated formal theory that proves
> > certain basic arithmetic truths, there is an arithmetical statement
> > that is true,[1] but not provable in the theory" (Kleene 1967, p.
> > 250).
>
> >http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
>
> > Is this a popularization? Many textbooks state the theorem in terms of
> > truth and provability.
>
> There is the syntactic version and the semantic version.
>
> The semantic version is "Given any primitive recursively axiomatisable
> theory T in a language in which the first-order language of arithmetic
> can be interpreted, such that T is arithmetically sound, there exist
> true sentences in the first-order language of arithmetic not provable
> in T." Goedel informally argues for this conclusion in the
> introduction to the paper.

I had the impression that Goedel argued this for a particular theory
not for any theory.

Anyway as stated above it is false unless by "first-order language of
arithmetic" you mean some restricted set of languages.

> The syntactic version is "Given any primitive recursively
> axiomatisable theory T in a language in which the first-order language
> of arithmetic can be interpreted, such that all primitive recursive
> functions can be represented in T and T is omega-consistent, there
> exist sentences in the first-order langauge of arithmetic which are
> independent of T." This is the version argued for in detail in the
> main text of the paper.> I do not know how Kleene would prove that in ANY conceivable theory
> > capable of arithmetic with a plausible interpretation there are
> > unprovable truths. (Although I tend to believe that "there is a
> > proposition G such that neither G nor not-G can be proven" holds for
> > ANY theory capable of arithmetic.)> >     but Goedel didn't rely on the notion
> > > >     of truth
>
> > > >     now because Goedel didn't rely on the notion
> > > >     of truth he cant tell us what true statements are
> > > >     thus his theorem is meaningless
>
> > > --
> > > ---------------------------
> > > |  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 videtur.
> > > |  BBB    aa a  r     bbb   |  
> > > ------------------------------ Hide quoted text -
>
> > > - Show quoted text -- Hide quoted text -
>
> > > - Show quoted text -- Hide quoted text -
>
> > > - Show quoted text -

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