New essay on Goedel's Incompleteness Theorem
in [Logic]
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From: Aatu Koskensilta on 27 Sep 2009 09:47 Scott H <zinites_page(a)yahoo.com> writes: > I still say that Goedel's theorem is founded on endless reference, as > t *is* a sentence in the model G is about. Here t is a term, not a sentence "in the model G is about", whatever that means. I'm afraid your further elucidations are of little help. As noted, the G�del sentence of a theory T, in the language of primitive recursive arithmetic, has the form (x)P(t,x) where t is a term the value of which is the code for G. What does it mean to say that "the truth value of G could turn out to be independent of its statement of reference, G'"? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 27 Sep 2009 09:48 David C. Ullrich <dullrich(a)sprynet.com> writes: > Aatu may be a real mathematician, but _you've_ been essentially > ignoring his attempts to "do his job". As a mathematician you should know very well logicians aren't real mathematicians. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 27 Sep 2009 09:55 Newberry <newberryxy(a)gmail.com> writes: > Let me add a few comments. I was booted out of here when I suggested > that Goedel's sentence was self-referential. Then I was lectured that > it is self-referential but not "literally self-referential." It > remainds me of Scotts theory that it acts like self-referential when > omega-consistency and recursive axiomatizability are added. I have no idea what you're on about. > I do not know how relevant these subtle distinctions are. It seems to > me that Goedel's sentence is basically self-referential. The G�del sentence G of a theory T is self-referential in a perfectly clear sense: it has the form (x)(P(x) --> Q(x)), where, provably in T, the only natural of which P holds is the code for G itself. Whether it's "literally self-referential" is a matter of taste. For didactic reasons people often stress that the G�del sentence of a theory is only about naturals, addition, and so on, to dispel any mistaken idea the sentence involves some peculiar non-arithmetical devices and what not. Of course, on this line of thought, the formalisation of the fundamental theorem of arithmetic is not "literally" about prime factorisations, the formalisation of the theorem that n^p * n^k = n^(p+k) is not "literally" about exponentiation, and so forth. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frederick Williams on 27 Sep 2009 10:23 Scott H wrote: > > On Sep 26, 10:51 am, Frederick Williams > <frederick.willia...(a)tesco.net> wrote: > > Your essay is no good though. > > Why? Because none of your friends think it is? I'm not aware of any friends of mine having read it. -- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested.
From: Newberry on 27 Sep 2009 13:44
On Sep 27, 6:55 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > Let me add a few comments. I was booted out of here when I suggested > > that Goedel's sentence was self-referential. Then I was lectured that > > it is self-referential but not "literally self-referential." It > > remainds me of Scotts theory that it acts like self-referential when > > omega-consistency and recursive axiomatizability are added. > > I have no idea what you're on about. > > > I do not know how relevant these subtle distinctions are. It seems to > > me that Goedel's sentence is basically self-referential. > > The Gödel sentence G of a theory T is self-referential in a perfectly > clear sense: it has the form (x)(P(x) --> Q(x)), where, provably in T, > the only natural of which P holds is the code for G itself. I am more familiar with this form ~(Ex)(Ey)(Pxy & Qy) (G) where Pxy means that x is a proof od y, and the only natural of which Q holds is the code for G itself. > Whether it's > "literally self-referential" is a matter of taste. For didactic reasons > people often stress that the Gödel sentence of a theory is only about > naturals, addition, and so on, to dispel any mistaken idea the sentence > involves some peculiar non-arithmetical devices and what not. Of course, > on this line of thought, the formalisation of the fundamental theorem of > arithmetic is not "literally" about prime factorisations, the > formalisation of the theorem that n^p * n^k = n^(p+k) is not "literally" > about exponentiation, and so forth. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon mann nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |