How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Aatu Koskensilta on 29 Jun 2010 09:27 MoeBlee <jazzmobe(a)hotmail.com> writes: > In the ordinary context, EVERY theory is infinite. Every theory is an > infinite set of sentences. So in a theory with no infinitary objects we need to adopt some more suitable representation for theories. A standard choice in case of PA, say, is to consider indices of Sigma-n sets of sentences for some n. -- 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 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 11:34
master1729 wrote: > CBL = ?? CBL /is/ Charlie-Boo's logic, even if it stands for something else. "Charlie-Boo's Logic" makes a nice mnemonic, actually. > wffs = ?? wff stands for "well-formed formula" Jim Burns |