How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
in [Logic]
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From: Aatu Koskensilta on 29 Jun 2010 09:29 herbzet <herbzet(a)gmail.com> writes: > Hope to hear a reply to you from someone who actually knows > what he's talking about. How did you like my reply? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Jun 2010 10:16 Charlie-Boo <shymathguy(a)gmail.com> writes: > Ok. What is the reference to the proof in ZFC of PA consistency doing > it that way? It's in Shelah's _Cardinal Arithmetic_ p.3245 - 4325238532. Basically, you just do a triple-fold transfinite recursion over a coherent extender sequence to obtain a suitable premouse, and iterate the upward Mostowski collapsing lemma a few times. To remove the extendible cardinal introduce some Aronszjan trees using Sacks forcing. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Jun 2010 10:22 Charlie-Boo <shymathguy(a)gmail.com> writes: > What is that formal expression? To find out you need to read a logic book. It appears the generous explanations various people have provided for your benefit in news are not sufficient. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: James Burns on 29 Jun 2010 12:20 Charlie-Boo wrote: > On Jun 29, 10:34 am, Chris Menzel > <cmen...(a)remove-this.tamu.edu> wrote: >>On Tue, 29 Jun 2010 03:34:12 -0700 (PDT), Charlie-Boo >><shymath...(a)gmail.com> said: >>>On Jun 29, 12:18 am, Chris Menzel >>><cmen...(a)remove-this.tamu.edu> wrote: >>>>The best known approach uses a mapping that Ackermann >>>>defined from the hereditarily finite sets into N >>> >>>There are too many sets to map them 1-to-1 with >>>the natural numbers. >> >>Apparently you have yet to master the semantic role >>of adjectives. > > Ok, then tell me. What is the semantic role of adjectives? Part of the semantic role of adjectives is to cause "hereditarily finite sets" to mean something different from "sets". Now that I've got you started, why don't you go ahead and see if you can figure the rest of it out. > http://blog.mrm.org/wp-content/uploads/2007/09/wizardofoz.jpg Who are you supposed to be? The Wizard? Dorothy? Toto? It can't be Toto, because Toto would have at least tried to figure out what the semantic role of adjectives was. Jim Burns
From: herbzet on 29 Jun 2010 22:33
Aatu Koskensilta wrote: > Frederick Williams writes: > > > Yes, you can: take Gentzen's proof (or Ackermann's etc) and formalize > > it in ZFC. > > This is a pretty silly way of proving the consistency of PA in set > theory. That PA is consistent is a triviality. The interest in Gentzen's > proof lies elsewhere. The axioms of PA are supposed to be reasonably self-evident truths about the naturals. But ... I read somewhere that seeing the truth of the infinite number of induction axioms of PA is in some sense equivalent to transfinite induction up to epsilon_0. That was a rather mysterious but intriguing remark to me. I wonder if it has anything to with what you're hinting at here. -- hz |