How Can ZFC/PA do much of Math - it Can't Even Prove PA isConsistent (EASY PROOF)
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From: MoeBlee on 30 Jun 2010 11:19 On Jun 30, 8:31 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > http://web.mat.bham.ac.uk/R.W.Kaye/papers/finitesettheory/ Thanks! MoeBlee
From: herbzet on 30 Jun 2010 17:54 Chris Menzel wrote: > herbzet said: > > Aatu Koskensilta wrote: > >> herbzet writes: > >> > >> > Hope to hear a reply to you from someone who actually knows > >> > what he's talking about. > >> > >> How did you like my reply? > > > > Fine, so far as it went. > > > > billh04 distinguished in his post between the consistency of > > PA being proved in ZFC, and a (meta) proof of the relative > > consistency of PA with regard to ZFC. > > > > You did not address the distinction drawn. > > What distinction do you have in mind beyond the fact that they are > simply two rather different proofs? Please clarify: by "simply two rather different proofs" do you mean "simply two rather different proofs of the same thing"? -- hz
From: herbzet on 30 Jun 2010 20:34 Chris Menzel wrote: > > On Tue, 29 Jun 2010 23:21:49 -0400, herbzet <herbzet(a)gmail.com> said: > > Aatu Koskensilta wrote: > >> herbzet writes: > >> > >> > Hope to hear a reply to you from someone who actually knows > >> > what he's talking about. > >> > >> How did you like my reply? > > > > Fine, so far as it went. > > > > billh04 distinguished in his post between the consistency of > > PA being proved in ZFC, and a (meta) proof of the relative > > consistency of PA with regard to ZFC. > > > > You did not address the distinction drawn. > > What distinction do you have in mind beyond the fact that they are > simply two rather different proofs? Well, let's put it this way: since the second proof is (I assume) itself formalizable in ZFC, are the two proofs not tantamount to the same thing? If we take ZFC to be sound, does not either proof suffice to establish that PA is consistent? -- hz
From: Charlie-Boo on 1 Jul 2010 00:30 On Jun 29, 9:54 pm, herbzet <herb...(a)gmail.com> wrote: > Chris Menzel wrote: > > billh04 said: > > > I thought that the proof of consistency of PA relative to ZFC by > > > showing that there is a model of PA in ZFC was a metatheorem, not a > > > theorem stated in ZFC and proved using the axioms of ZFC. > > > You can formalize the metatheory for the language of PA in ZF and prove > > in ZF the existence of a model of PA and, hence, that PA is consistent, > > by the soundness theorem. Do you have a reference of anyone showing the entire proof that PA is consistent being carried out in ZFC? C-B > I take it that "a metatheorem" is a theorem *about* a formal theory T > rather than a theorem *of* T (although of course these are not mutually > exclusive categories). > > But what is a "metatheory", and specifically, what is "the metatheory > for the language of PA"? > > -- > hz
From: Charlie-Boo on 1 Jul 2010 01:56
On Jul 1, 12:31 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > Do you have a reference of anyone showing the entire proof that PA is > > consistent being carried out in ZFC? > > You desire to rest your eyes on a hideous jumble of unintelligible > formalities? What for? Why in the world would they be unintelligible? To show that it can be done, for one. Didn't you try that earlier - as in 14 minutes ago? "Why would you want to see it in the first place?" Is that supposed to be an answer to a question or a proof or a theorem or what?? C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |