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 5 Jul 2010 18:14 MoeBlee <jazzmobe(a)hotmail.com> writes: > On Jul 5, 7:05�am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: >> >> > Aatu said PA is consistent, _period_, without any formal proof? >> >> There seems to be some confusion over my (perfectly standard as always) >> take on these matters. > > Is there anything in my paraphrase several posts ago (in the full > context I gave it) that is inaccurate? Not that I can see. It seems that it has given an incorrect impression of what I think about these matters to Nam, though. I took your paraphrase to be simply an explanation of what I mean when I say I don't regard the usual consistency proofs for PA as establishing any "relative" sort of consistency -- that is, I took you to be explaining I were asserting PA does not prove P and not-P for any sentence P. and not merely ZF proves "PA does not prove P and not-P for any sentence P". or It is provable in ZF that PA does not prove P and not-P for any sentence P. which is indeed what I intended. > However that quote is to be understood, for the record, I did not say > that you hold there is no formal proof that PA is consistent, but > rather that you hold PA is consistent on (for lack of better term I > can think of right now) even more basic grounds than formal proof. Right. I hold PA is consistent because there's a mathematical proof of this fact, that is, a piece of compelling mathematical reasoning establishing the claim, invoking only principles we usually take for granted when doing mathematics. I can't really see how anyone who didn't take such things for granted could conclude from the "relative consistency" theorems anything about the consistency of PA. > And that is what I have highlighted as to your view. It seems your posts have nevertheless led Nam and lwalke to incorrect ideas about my views. This is not any fault of yours. -- 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 7 Jul 2010 13:27 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: > >> 1. We give an inductive definition of a property T(x) of >> arithmetical sentences. > > So, would T(x) be a formula in L(PA)? No. -- 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 7 Jul 2010 13:29 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > MoeBlee wrote: > >> And, in that regard, I have said all along that there is no >> finitistic proof of the consistency of PA. > > So you've agreed that there's no formal proof for PA's consistency and that > if you go by formal proof only then you don't have knowledge of PA's > consistency. There are many formal proofs of the consistency of PA. None of them are finitistic. -- 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 7 Jul 2010 13:44 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > I mean, if we don't actually know something isn't kind of prudent that > we just say we don't know that something? (And I was blamed for being > a crank?) What we take to be known in mathematics is to some extent a matter of opinion. In ordinary mathematics it's a triviality that PA is provably consistent. Those who regard the principles and modes of reasoning taken for granted in ordinary mathematics as dubious will of course have to inspect the proof for themselves and make up their own mind. -- 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 7 Jul 2010 13:45
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Huh? You didn't say anything in that conversation and silently let him > "represent" your views and if his "representing" goes wrong that would > be my fault, no explanation needed? I said nothing about anything being your fault. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |