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 08:23 Charlie-Boo <shymathguy(a)gmail.com> writes: > What of ZFC's set theoretic axioms is necessary - especially not > bookkeeping ones like sets existing that are used throughout PA and > are not needed in every proof? Various bookkeeping axioms about set existence are used in the proof. Of course, for the consistency of PA ZFC is a huge overkill. We can prove PA is consistent in e.g. ACA, using a (definable) truth predicate for arithmetical statements. -- 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 08:28 George Greene <greeneg(a)email.unc.edu> writes: > I really don't think that the model existence theorem is going to leap > out at him here. Pretty much any elementary text contains an account of the set theoretic construction of the various number systems. It's a trivial exercise to verify the axioms of PA hold in the structure of naturals. Combined with the soundness of first-order logic this immediately yields the consistency of PA. -- 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 08:29 Charlie-Boo <shymathguy(a)gmail.com> writes: > No no no. You have to prove it using ZFC's axioms and rules only. No no no. You have to prove it to Charlie-Boo's satisfaction, with broccoli sticking out of your ears, drunk on too much cheap whiskey. -- 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 08:34 Transfer Principle <lwalke3(a)lausd.net> writes: > 1. Is there a proof in ZF-Inf+~Inf that every set is finite? It appears I once posted a proof. It relies, in an essential way, if I recall correctly, on the axiom of replacement. -- 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 08:58
Charlie-Boo <shymathguy(a)gmail.com> writes: > How about when you said that Gentzen proved PA consistent using ZFC? No one ever proves anything using ZFC in the sense of producing formal derivations. Rather, they prove mathematical theorems using mathematical principles formalized in ZFC. By certain elementary considerations we know the formalizations of such theorems are formally derivable in ZFC. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |