New essay on Goedel's Incompleteness Theorem
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From: Scott H on 28 Sep 2009 21:37 On Sep 28, 9:14 pm, I wrote: > to the detriment these newsgroups. That should be 'of these newsgroups.' I hope I don't have to hear voices over it tonight.
From: David C. Ullrich on 29 Sep 2009 06:26 On Mon, 28 Sep 2009 16:32:24 +0300, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: >David C. Ullrich <dullrich(a)sprynet.com> writes: > >> For heaven's sake. If T and T* have the same axioms then they're the >> same theory. > >Nam will no doubt soon chime in and point out we are, in comments like >above, implicitly admitting something he was adamantly insisting on a >while back... Heh-heh. Good point. Let's revise this: For heaven's sake, if T and T* are theories in the same formal system with the same axioms then they're the same theory. David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
From: Frederick Williams on 29 Sep 2009 06:35 Scott H wrote: > > ... I will look elsewhere for > discussion, ... Promises, promises. -- 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: Scott H on 29 Sep 2009 07:18 On Sep 28, 9:24 am, David C. Ullrich <dullr...(a)sprynet.com> wrote: > On Sun, 27 Sep 2009 14:09:53 -0700 (PDT), Scott H > <zinites_p...(a)yahoo.com> wrote: > >If we have a nest of theories > > >T > T' > T'' > T''' ... > > >all with the same axioms, > > For heaven's sake. If T and T* have the same axioms then > they're the same theory. But T' is *contained* in T; it is not *equal* to T. Granted, they *might* be the same theory. This, I believe, is a question for Platonists. In the section 'On the Possible Existence of Supernatural Numbers,' I describe how T can be extended by adding a proof of G' that is inductively inaccessible from outside T, but becomes accessible within T by functioning like a variable. I hope that this idea will make Goedel's theorem and omega-inconsistency easier for people to understand. On a side note, suppose you defined a truth predicate in T and formulated a statement similar to the Liar Statement by reference to its reflection in T': L = [~Tr [~Tr S x]] <=> [~Tr [~Tr [~Tr S x]]] <=> [~Tr L']. This would give us a truth value of L' that differed from that of L, would it not? Ultimately, we'd end up with the sequence of truth values T -> F -> T -> F ..., or F -> T -> F -> T ... for what were thought to be 'the same statement' in the nested theories T, T', T'' ... Correct me if I'm wrong, but I think this may be the basis of Tarski's theorem.
From: Daryl McCullough on 29 Sep 2009 08:22
Scott H says... >On a side note, suppose you defined a truth predicate in T and >formulated a statement similar to the Liar Statement by reference to >its reflection in T': > >L = [~Tr [~Tr S x]] ><=> [~Tr [~Tr [~Tr S x]]] ><=> [~Tr L']. > >This would give us a truth value of L' that differed from that of L, >would it not? If Tr(x) is any formula of first-order logic, then you can come up with a formula L such that L <-> ~Tr(#L) where #L means the Godel number of L. There is no L' involved. Yes, if Tr(x) were a truth predicate, this would be a contradiction, which is an argument that there is no truth predicate for arithmetic definable in the language of arithmetic. -- Daryl McCullough Ithaca, NY |