How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent(EASY PROOF)
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From: Nam Nguyen on 30 Jun 2010 02:45 R. Srinivasan wrote: > On Jun 30, 9:09 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> The answer imho is simple: they, the "standard theorists" (and I use >> the phrase in a respectful way), would assert they somehow "know" >> the natural numbers and this "standard model for the LANGUAGE of PA" >> is just the natural numbers, collectively! >> > And from this lofty platform of "rigor", they denounce any dissent as > "ignorant confusion". Many times people would fight to the bitter end for no apparent reasons other than that's just the way they've been brought up and taught as "right". > Looks to me like the cancer of infinity is so deeply ingrained in the > thinking of classical logicians that they are incapable of > appreciating any attempt to remove this cancer from logic. Imho, "cancer of infinity" is too strong a word (though I think I know what you meant by it). No mathematics would be worthwhile without addressing the "issue" of infinity. And this is where the "standard theorists" and the "relativists" would fight: to the former the "issue" is swept under the rug of "Induction" with _no problem_, while to the later the sweeping has the consequence that there would be anti-induction unknowable we have to _accept and formalize_! (Iow, there's no free lunch, so to speak, in dealing with the issue of infinity). > There are absolute (Platonic) truths in NAFL (e.g. a NAFL theory is > either consistent or inconsistent) but such propositions are not > formalizable in the language of a NAFL theory. These must remain as > metamathematical truths. But isn't inconsistency first order provable (hence formalizble), which is different from consistency?
From: Frederick Williams on 30 Jun 2010 11:16 Nam Nguyen wrote: > The answer imho is simple: they, the "standard theorists" (and I use > the phrase in a respectful way), would assert they somehow "know" > the natural numbers and this "standard model for the LANGUAGE of PA" > is just the natural numbers, collectively! You should read 'What numbers could not be' in Benacerraf and Putnam. Johnny is von Neumann, Ernie is Zermelo. -- I can't go on, I'll go on.
From: Nam Nguyen on 1 Jul 2010 10:01 MoeBlee wrote: > On Jun 29, 10:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Aatu Koskensilta wrote: >>> Frederick Williams <frederick.willia...(a)tesco.net> writes: >>>> Yes, you can: take Gentzen's proof (or Ackermann's etc) and formalize >>>> it in ZFC. >>> This is a pretty silly way of proving the consistency of PA in set >>> theory. >>> That PA is consistent is a triviality. >> In what formal system is this triviality in? > > It's a theory of Z-R, for example. Whether it's "trivial" to prove in > Z-R depends on what strikes one as trivial. > >> (Iow, you didn't mean >> it's a fact that PA is syntactically consistent, did you?) > > Consistent IS syntactically consistent. But there's also such thing as relative consistency proof! For example, from T = {Ax[xex] /\ ~Ax[xex]}, it's a triviality to prove the consistency of PA, but should I proclaim that PA is consistent, as in, "that PA is consistent is a triviality", as Aatu put it? The question I had for him was a clarification request to see if he meant PA is really consistent, or if he meant that was just a relative consistency proof he had referred to. (You should read people's conversation more carefully, before jumping to conclusion whether or not people understand this or that.) > > Here's one among equivalent definitions: > > DEFINITION OF CONSISTENT: > > A set of formulas S is in a language is consistent iff there is no > formula P and the negation of P in S. > > PERIOD. I was about to ignore your incorrectness here, but the tallness of your ending "PERIOD" seemed defying any, say, "forgiveness". So here it is. Let GC, cGC be 2 1st order formulas: GC <-> Goldbach Conjectures cGC <-> "There are infinitely many counter examples of GC" Let S be a set of formulas in L(PA) and be defined as S = {GC, cGC}. So the 2-formula in S _conforms precisely_ with your "DEFINITION OF CONSISTENT" (they aren't negations of each other as you clearly demanded) and yet _nobody but you_ would refer to S as a set about formal system consistency! Perhaps you'd want to review basic textbooks for a precise definition of a formal system consistency? (Or at least keep the "lecturing" tone to yourself!) > > That a set of FIRST order formulas is consistent iff that set of > sentences is satisfiable is a RESULT we prove. > > And, of course, Aatu is claiming that PA is consistent. Since he hasn't responded to my original question, I'm not going to comment about _his_ "claim". > He's been > saying it for at least about a decade. What don't you understand about > that? Can you disprove a formula in a T, using only syntactical means (i.e. rules of inference and T's axioms)? Can you define the naturals using only the definition of language model, and _without being circular_ ? What I don't understand is why there are those, and you're one of them, who'd keep proclaiming PA is consistent while not being able to satisfactorily answer "Yes" to the 2 questions above.
From: Nam Nguyen on 1 Jul 2010 10:34 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> but should I proclaim that PA is consistent, as in, "that PA is >> consistent is a triviality", as Aatu put it? > > You will of course decide for yourself what you wish to proclaim. But it's _your_ proclamation that precipitates this conversation, not mine. So apparently in your view anybody, including yourself, could state anything one would "wish to proclaim"! Even a relativist wouldn't go that far! > >> The question I had for him was a clarification request to see if he >> meant PA is really consistent, or if he meant that was just a relative >> consistency proof he had referred to. > > The trivial consistency proof for PA is no more relative than any other > proof in mathematics. I'd prefer a straightforward kind of answer that this is a relative proof of (PA's) consistency in another formal system, ZFC, ZF-R,..., rather than a less straightforward kind answer like "no more relative". This is after all reasoning, which should be precise and clear-cut: if it's a relative proof then it should be so stated, at least upon a clarification request! > Since you apparently take the view that pretty > much everything in mathematics is relative you will naturally regard the > proof as relative, What did you mean by "pretty much" here? Given FOL frame work, I've never claimed the proof of Ax[~(Sx=0)] is relative in PA! You seem to have over- exaggerated my view. No? > just as you will regard Dirilecht's theorem, the > deduction theorem, the Beurling-Lax theorem, and so on. Concentrating on > trivial consistency proofs is totally arbitrary. Again this is just an over-exaggeration or misunderstanding of my view.
From: Nam Nguyen on 1 Jul 2010 11:15
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> I'd prefer a straightforward kind of answer that this is a relative >> proof of (PA's) consistency in another formal system, ZFC, ZF-R,..., >> rather than a less straightforward kind answer like "no more >> relative". > > We have in mathematics results such as > > If the generalized Riemann-Hypothesis holds, the deterministic > Miller-Rabin primality test runs in polynomial time. > > If ZFC + "there are infinitely many Woodin cardinals" is consistent, > so is ZF + AD + DC. > > and so on, which are naturally regarded as relative, in that they state > that something holds relative to the assumption that some as yet > unproven statement holds. The theorem that PA is consistent, that 4 + 4 > = 8, Dirilecht's theorem, and so on, are not relative in this ordinary > sense. I don't know of Dirilecht's theorem, but I know why the proof 4+4=8 in PA isn't relative: we can _syntactically_ use rules of inference to prove it, _even if PA itself is inconsistent_! But that leaves the consistency of PA: by what reason would we proclaim it be NOT relative, given that on the surface it's not in the same category theorem as 4+4=8 in PA, in the sense nobody can prove a disproof using rules of inference. > We will of course regard them as relative if we consider the > principles usually accepted in ordinary mathematics as conjectural or > doubtful. It's not about doubting certain things: some of the statements in FOL reasoning are above and beyond doubt-or-not-doubt. And they're relative statements and should be accepted for what they really are. > Even so concentrating on consistency results is totally > arbitrary. Of course GIT would need such arbitrariness! |