New essay on Goedel's Incompleteness Theorem
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From: Aatu Koskensilta on 21 Sep 2009 13:24 "Scott H" <nospam> writes: > I give a concise, formal proof and a short intuitive description of ZFC + > ~G. > > http://www.hoge-essays.com/incompleteness.html > > Any constructive feedback is welcome. Well, there's no formal proof in your essay, and I don't see any mention of ZFC. My constructive feedback consists mainly of the suggestion that, should you for some reason take interest in the incompleteness theorems, you go through a good text on the subject. Your musings about "endless reference", as invoked in passages such as We must remember, however, that G�del's theorem is founded not on self-reference but on endless reference, and that the truth value of G could turn out to be independent of the truth value of its statement of reference, G'. for example appear to be based on some confusion the nature of which eludes me, and which you could probably profitably sort out for yourself by working through the mathematical details. The stuff about supernatural numbers and what not seems also confused -- a case in point, unlike you imply there's no mystery to induction for first-order properties holding in a non-standard model. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: LauLuna on 21 Sep 2009 14:57 On Sep 21, 4:33 pm, "Scott H" <nospam> wrote: > I give a concise, formal proof and a short intuitive description of ZFC + > ~G. > > http://www.hoge-essays.com/incompleteness.html > > Any constructive feedback is welcome. Why do you think there is an endless reference in Gödel's sentence? Regards
From: Scott H on 21 Sep 2009 21:56 On Sep 21, 1:24 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Well, there's no formal proof in your essay, and I don't see any mention > of ZFC. I do mention Zermelo-Fraenkel set theory at one point, and the comments about supernatural numbers in PA also apply to ZFC. > My constructive feedback consists mainly of the suggestion that, > should you for some reason take interest in the incompleteness theorems, > you go through a good text on the subject. My proof is based on the one in Goedel's original manuscript. I'd be interested to hear where you thought my confusion lay.
From: Scott H on 21 Sep 2009 22:03 On Sep 21, 2:57 pm, LauLuna <laureanol...(a)yahoo.es> wrote: > Why do you think there is an endless reference in Gödel's sentence? I show this near the middle of the section 'How Endless Propositional Reference Leads to Unanswered Questions About Numbers.' The substitution or arithmoquine function, which replaces the free variable of a property with the symbol for that property, allows an infinite substitution process to take place. Ultimately, we obtain a statement similar to: The following is unprovable: The following is unprovable: The following is unprovable: ... It is through the concepts of omega-consistency and recursive axiomatizability that the provability of the first statement becomes logically connected with the provability of the second.
From: Newberry on 21 Sep 2009 23:06
On Sep 21, 7:03 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > On Sep 21, 2:57 pm, LauLuna <laureanol...(a)yahoo.es> wrote: > > > Why do you think there is an endless reference in Gödel's sentence? > > I show this near the middle of the section 'How Endless Propositional > Reference Leads to Unanswered Questions About Numbers.' > > The substitution or arithmoquine function, which replaces the free > variable of a property with the symbol for that property, allows an > infinite substitution process to take place. Ultimately, we obtain a > statement similar to: > > The following is unprovable: The following is unprovable: The > following is unprovable: ... Where is the self-reference in this? > It is through the concepts of omega-consistency and recursive > axiomatizability that the provability of the first statement becomes > logically connected with the provability of the second. |