How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Alan Smaill on 3 Jul 2010 14:38 Charlie-Boo <shymathguy(a)gmail.com> writes: > On Jul 3, 2:28�pm, Charlie-Boo <shymath...(a)gmail.com> wrote: >> On Jul 3, 2:05�pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> >> > Charlie-Boo <shymath...(a)gmail.com> writes: >> > > And you won't post it, alas, so people could quickly and easily see it >> > > and debunk it like the other BS references. >> >> > You want me to post Ross Bryant's Master's Thesis? You'll find it online >> > at: >> >> > �http://www.cas.unt.edu/~rdb0003/thesis/thesis.pdf >> >> Thanks. �But where does he show proof lines with ZFC as their >> justification? �He refers to ZFC and makes claims regarding it, but >> all of his proofs and arguments are presented in normal mathematical >> terms with no reference to ZFC. >> >> 100 pages full of proofs and a handful of references to ZFC doesn't do >> it. > > And the proof that it can't be carried out in PA? CBL worked that out years ago. >> C-B > -- Alan Smaill
From: Jesse F. Hughes on 4 Jul 2010 08:59 Transfer Principle <lwalke3(a)lausd.net> writes: > If Aatu can say that PA is consistent, _period_, without any formal > proof whatsoever, then why can't Nguyen believe that PA is > inconsistent, _period_, without formal proof? Aatu said PA is consistent, _period_, without any formal proof? -- Jesse F. Hughes "If you believe there is any other truth but what is in your mind, you are deluding yourself." -- Demers Paradox
From: Aatu Koskensilta on 5 Jul 2010 09:40 Charlie-Boo <shymathguy(a)gmail.com> writes: > And the proof that it can't be carried out in PA? How would you express Borel determinacy in the first-order language of arithmetic? -- 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 5 Jul 2010 10:05 "Jesse F. Hughes" <jesse(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. Allow me to clarify. I know that PA is consistent because I know how to produce a mathematical argument for that fact which I fully understand -- in the sense that I grasp the key idea and can explain all the technical details that go into its execution -- and find compelling. The proof I have in mind has the following structure: 1. We give an inductive definition of a property T(x) of arithmetical sentences. 2. We show that T("P") iff not T("~P"), that T("P & Q") iff T("P") and T("Q"), and so on. 3. We conclude from this that T does not hold of any contradiction. 4. We show that T holds of all the axioms of PA. 5. We show that the rules of inference of first-order logic preserve T. 6. We conclude that no contradiction is formally derivable in PA. We can of course formalize this proof in any number of theories -- ACA, ZFC, ... -- but this is just an incidental technical observation of no immediate interest as far as consistency of PA is concerned. We may also be interested in knowing exactly which set theoretic axioms are needed e.g. to justify the sort of inductive definitions involved, and so on. Again, this has no immediate connection to the question of consistency of PA. As with any proof in ordinary mathematics we take in the above proof many mathematical principles and modes of reasoning for granted. For example, we apply induction to a property involving a predicate defined by means of an infinitary inductive definition. Everyone must decide for themselves whether they find such invocations evident, compelling, legitimate mathematics. It's a brute fact of life that in ordinary mathematics we do in fact take them for granted. In Usenet debates, and elsewhere too, people like to make much of and pontificate tediously on who has or does not have the "burden of proof". In reality, such "logical" rules of formal debate have about as much to do with real arguments, persuasion, conversion, as the rules of cricket. Whoever wants to convince someone else of something must of course present arguments, questions, reflections, examples, illustrations that have some real force to the receiving person, regardless of whether they make a "positive claim" according to some rulebook of debating. I don't have any real interest in convincing anyone who does not take ordinary mathematics for granted of much anything. I'm content with simply noting that the consistency of PA is a triviality given what we in fact usually take for granted in mathematics. If someone tells me they find the usual modes of reasoning horribly dubious or unclear, even after careful explanation and meticulous elucidation, I simply shrug and move on. -- 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 5 Jul 2010 18:08
MoeBlee <jazzmobe(a)hotmail.com> writes: > And, in that regard, I have said all along that there is no finitistic > proof of the consistency of PA. I base that on the second > incompleteness theorem. The first incompleteness theorem already suffices. This elementary point is often overlooked, and has recently been stressed by Richard Zach. I said a bit about this in an earlier post, not that it matters much in the grand scheme of things. > (I haven't personally verified every detail in a proof of the second > incompleteness theorem, so my remarks are to the extent we can be > confident that the second incompleteness theorem does indeed withstand > complete scrutiny, as it is reported in the literature that it does.) Come now, you're surely not hedging your bets and saying you're not absolutely certain the second incompleteness theorem, which has withstood the most exacting and extreme scrutiny humanly possible, far, far beyond what any run of the mill result in mathematics -- the classification of finite groups, the Heine-Borel theorem, most of stuff in analysis, Wiles's proof of Fermat's last theorem, the fundamental theorem of arithmetic ... -- usually has to endure to prove its good name, holds. (I apologise for the clumsiness in the construction of the previous sentence. My only excuse is that I'm again suffering of severe lack of sleep.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |