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 09:13 Charlie-Boo <shymathguy(a)gmail.com> writes: > ZFC was designed to avoid paradoxes by making explicit what can be a > set. It doesn't do anything else except what the Peano Axioms give > it. Does PA give us Borel determinacy? Is Borel determinacy trivial? -- 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 09:14 Charlie-Boo <shymathguy(a)gmail.com> writes: > On Jun 28, 8:58�am, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > >> Yes, you can: take Gentzen's proof (or Ackermann's etc) and formalize >> it in ZFC. > > Give the slightest bit of details. Read a logic book. >> It has everything to do with V_omega. > > That's not a ZFC axiom. Indeed. It is a set. -- 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 09:16 billh04 <hale(a)tulane.edu> writes: > Are you saying that it is a theorem of ZFC that PA is consistent? Sure. That is, the statement "PA is consistent" formalized in the language of set theory as usual is formally derivable in ZFC (and already in much weaker theories). -- 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 09:18 MoeBlee <jazzmobe(a)hotmail.com> writes: > By the way, here's Aatu's proof: > > Theorem (of Z-"ax regularity"-"ax infinity"+"~ ax infinity"): > > Ax x is finite. You need an F up there, I believe. -- 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 09:27
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 |