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From: Rupert on 22 Jun 2010 01:04
On Jun 22, 1:52 pm, Newberry <newberr...(a)gmail.com> wrote:
> 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.
>

If I recall correctly he argued it for the formal system of "Principia
Mathematica" (with the axiom of infinity added) and then went on to
indicate how it generalised to other formal systems.

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

The first-order language of arithmetic just means the first-order
language with constant symbol 0, unary function symbol s (for
successor), and binary function symbols + and *. And when I speak of a
theory as being "arithmetically sound" or a sentence as being "true",
I have in mind the standard model. You need to have the concept of the
standard model if you are going to make sense of the semantic version
of the incompleteness theorem.

The semantic version of the incompleteness theorem can certainly be
proved in ATR_0, for example, so if you think that it is false then
that must mean that you do not accept that theory. You should make it
clear what basic assumptions you are working with.
From: Newberry on 22 Jun 2010 11:38
On Jun 21, 10:04 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Jun 22, 1:52 pm, Newberry <newberr...(a)gmail.com> wrote:
>
> > 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.
>
> If I recall correctly he argued it for the formal system of "Principia
> Mathematica" (with the axiom of infinity added) and then went on to
> indicate how it generalised to other formal systems.
>
> > Anyway as stated above it is false unless by "first-order language of
> > arithmetic" you mean some restricted set of languages.
>
> The first-order language of arithmetic just means the first-order
> language with constant symbol 0, unary function symbol s (for
> successor), and binary function symbols + and *. And when I speak of a
> theory as being "arithmetically sound" or a sentence as being "true",
> I have in mind the standard model. You need to have the concept of the
> standard model if you are going to make sense of the semantic version
> of the incompleteness theorem.
>
> The semantic version of the incompleteness theorem can certainly be
> proved in ATR_0, for example, so if you think that it is false then
> that must mean that you do not accept that theory. You should make it
> clear what basic assumptions you are working with.

Fair enough. I am saying that there is a theory capable of arithmetic
as we know it with a plausible interpretation such that all the
unprovable sentences are not true.

I think it is the folks who state that there are true but unprovable
sentences who do not clearly state what the assumptions are.

And yes, I do think that the separation of truth and provability is
paradoxical. I know this statement is not going to win me any friends
or influence people but I had to tell you the truth.
From: George Greene on 22 Jun 2010 14:16
On Jun 21, 12:10 am, Newberry <newberr...(a)gmail.com> wrote:

> 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).

Unfortunately, Newberry continues,
> I do not know how Kleene would prove that in ANY conceivable theory
> capable of arithmetic with a plausible interpretation there are
> unprovable truths.

KLeene would NOT prove that, since Kleene DID NOT SAY that.
What Kleene DID say was that for any EFFECTIVELY GENERATED formal
theory,
there were unprovable (i.e. UN-effectively-generatable) truths.
He did NOT say that this was for "any conceiveable" theory.


Well, I hope that clears THAT up.

Of course, there are those of us who think that "un-effectively-
generatable formal theory"
IS A CONTRADICTION IN TERMS (what MAKES a theory "formal" is that you
CAN
generate it), but standard technical parlance is that ANY old "full"
set of consequences
(even if there is no effective way of filling it out) is a theory.
"True" "natural" first-order arithmetic is such a "theory".
From: Rupert on 22 Jun 2010 20:49
On Jun 23, 1:38 am, Newberry <newberr...(a)gmail.com> wrote:
> On Jun 21, 10:04 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
>
> > On Jun 22, 1:52 pm, Newberry <newberr...(a)gmail.com> wrote:
>
> > > 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.
>
> > If I recall correctly he argued it for the formal system of "Principia
> > Mathematica" (with the axiom of infinity added) and then went on to
> > indicate how it generalised to other formal systems.
>
> > > Anyway as stated above it is false unless by "first-order language of
> > > arithmetic" you mean some restricted set of languages.
>
> > The first-order language of arithmetic just means the first-order
> > language with constant symbol 0, unary function symbol s (for
> > successor), and binary function symbols + and *. And when I speak of a
> > theory as being "arithmetically sound" or a sentence as being "true",
> > I have in mind the standard model. You need to have the concept of the
> > standard model if you are going to make sense of the semantic version
> > of the incompleteness theorem.
>
> > The semantic version of the incompleteness theorem can certainly be
> > proved in ATR_0, for example, so if you think that it is false then
> > that must mean that you do not accept that theory. You should make it
> > clear what basic assumptions you are working with.
>
> Fair enough. I am saying that there is a theory capable of arithmetic
> as we know it with a plausible interpretation such that all the
> unprovable sentences are not true.
>

Well, feel free to elaborate.

> I think it is the folks who state that there are true but unprovable
> sentences who do not clearly state what the assumptions are.
>

Well, you're wrong about that. The underlying assumptions are made
absolutely clear and precise.

> And yes, I do think that the separation of truth and provability is
> paradoxical. I know this statement is not going to win me any friends
> or influence people but I had to tell you the truth.

Well, that's fine, but as I say, you need to clarify what your own
underlying assumptions are.
From: Newberry on 23 Jun 2010 00:44
On Jun 22, 5:49 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Jun 23, 1:38 am, Newberry <newberr...(a)gmail.com> wrote:
>
> > On Jun 21, 10:04 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
>
> > > On Jun 22, 1:52 pm, Newberry <newberr...(a)gmail.com> wrote:
>
> > > > 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.
>
> > > If I recall correctly he argued it for the formal system of "Principia
> > > Mathematica" (with the axiom of infinity added) and then went on to
> > > indicate how it generalised to other formal systems.
>
> > > > Anyway as stated above it is false unless by "first-order language of
> > > > arithmetic" you mean some restricted set of languages.
>
> > > The first-order language of arithmetic just means the first-order
> > > language with constant symbol 0, unary function symbol s (for
> > > successor), and binary function symbols + and *. And when I speak of a
> > > theory as being "arithmetically sound" or a sentence as being "true",
> > > I have in mind the standard model. You need to have the concept of the
> > > standard model if you are going to make sense of the semantic version
> > > of the incompleteness theorem.
>
> > > The semantic version of the incompleteness theorem can certainly be
> > > proved in ATR_0, for example, so if you think that it is false then
> > > that must mean that you do not accept that theory. You should make it
> > > clear what basic assumptions you are working with.
>
> > Fair enough. I am saying that there is a theory capable of arithmetic
> > as we know it with a plausible interpretation such that all the
> > unprovable sentences are not true.
>
> Well, feel free to elaborate.

http://www.scribd.com/doc/26833131/When-Are-Relations-Neither-True-Nor-False
>
> > I think it is the folks who state that there are true but unprovable
> > sentences who do not clearly state what the assumptions are.
>
> Well, you're wrong about that. The underlying assumptions are made
> absolutely clear and precise.

Considering again this quote from Wkipedia

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).

Where are the underlying assumptions with respect to truth made
absolutely clear and precise?

>
> > And yes, I do think that the separation of truth and provability is
> > paradoxical. I know this statement is not going to win me any friends
> > or influence people but I had to tell you the truth.
>
> Well, that's fine, but as I say, you need to clarify what your own
> underlying assumptions are.

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