How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: MoeBlee on 3 Jul 2010 16:27 On Jul 2, 8:21 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 1, 9:25 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jul 1, 9:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > or if he meant that was just a relative consistency > > > proof he had referred to. > > Anyway, Aatu is not saying just that there exists a relative > > consistency proof nor just that, say, ZF or some other formal system > > proves Con(PA), but rather he's saying that PA IS consistent. He's > > saying that aside from whatever FORMAL proofs, PA is consistent - > > PERIOD. His basis is for that is not a FORMAL proof, but rather his > > conviction that the axioms of PA are true (and not even in confined to > > a FORMAL model theoretic sense of truth, but rather that the axioms > > are simply true about the natural numbers, as we (editorial 'we') > > understand the natural numbers even aside from any formalization. > > But this raises an interesting point here. > > 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? Who's stopping him? However, Aatu does give his REASON for his pre- formal belief that PA is consistent. For that matter, why can't Herc > believe > that there exist only countably many reals, _period_, without formal > proof, or Srinivasan believe that Infinity is false, _period_, without > formal > proof, or WM believe that certain large naturals don't exist, > _period_, > without formal proof? Same answer as above. > In reality, the only statements that posters are allowed to make, > _period_, without formal proof, are those which ZFC proves, or at > least > are undecidable in ZFC. Any statement outright refuted by ZFC isn't > allowed to be made, _period_, without formal proof. I've never seen such a rule. > On the contrary, a > formal proof will be demanded at best, with the poster who made the > claim being asked to provide axioms, definitions, and so on. And at > worst, the poster will be called a five-letter insult. If a person is clear that they are adopting some mathematical belief on a pre-formal basis, then fine. And we may ask for the pre-formal reasons for adopting such a belief. That is different from the kind of ignorant, incoherent, and factually incorrect statements made variously by various cranks. > So Nguyen is on the right track here. Aatu can state that PA is > consistent because there's a theory T proves such -- even if Aatu > states it without any such formal proof -- and that theory is ZFC. Actually just Z-R is sufficient, and, as I understand, certain weaker theories too. > But > no one is allowed to make a claim of anything refuted by ZFC without > a formal proof in some (other) theory. This is a fact, no matter how > much MoeBlee or anyone else may try to deny it. You are simply ignorant. There are mathematicians who claim that the axiom of choice is false. (I can't provide you with names without re- researching, but you'll find such in the literature). Moreover, even I have said, in so many words, that if anyone believes any axiom of ZF is false, then I don't object to that IN ITSELF. For example, if someone believes the axiom of infinity is false then that doesn't bother me in the least. MoeBlee
From: MoeBlee on 3 Jul 2010 16:40 On Jul 2, 9:10 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > I agree with you in the above: there's a degree of being double > standard that Moeblee and other "standard theorists" How EXACTLY am I a "standard theorist"? > have exercised: > they'd blast people as talking nonsense if people don't conform to > standard logics No, anyone is free to propose an alternative logic and present arguments in it. > in arguing while when it's their turn to conform > to definitions of consistency and language models in proving, say, > PA's syntactical consistency, it'd be perfectly OK for them to > _ignore formal proofs and just use mere intuitions_! I never proposed that mere intuition is sufficient as proof. You're getting really close to lying about me. Moreover, I do usually conform to standard definitions. But some terms do have different definitions in the literature, and so one may choose from among them. Moreover, definitions are stipulative and often in context of one author's (or poster's) purposes in context. And, as to the recent definition of 'consistency', I even admitted that the first definition I gave is awkward and that the actual definition I use and would use subsequently is indeed a standard definition. > The most interesting question is why MoeBlee and those "standard theorists" > never admit they're just being philosophical about PA's consistency, What are you talking about? I've never claimed the consistency of PA on some philosophical basis. Nor have I claimed that proof of consistency of PA from, say Z set theory, carries any epistemological value. This is old news: I understand (per the second incompleteness theorem, modulo any details I have not personally confirmed) that there is no finitistic proof of the consistency of PA. You've built up a whole characterization of my philosophical notions but that characterization is not supported by anything I've actually posted. > while they blatantly admit that they've gone astray from rigorousness > of reasoning? I mean MoeBlee said above: > > "aside from whatever FORMAL proofs, PA is consistent". I didn't state that as MY view. I was telling you AATU's view. For Christ sakes! MoeBlee
From: MoeBlee on 3 Jul 2010 16:44 On Jul 3, 4:53 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > in all references given (Gentzen by Frederick Williams and > Hinman by MoeBlee) there is no mention of any of ZFCs axioms anywhere > in their proofs (including earlier results on which it relies) at > all. Hinman's proof concerns ZF (so ZFC a fortiori). Of course certain of the ZF axioms are used in Hinman's proof as that proof depends on certain results he references from earlier chapters. MoeBlee
From: Jack Campin - bogus address on 3 Jul 2010 17:20 >> What Charlie-Boo needs, therefore, is some evidence that >> convinces him that if one were to work out all the steps, one >> can eventually prove Con(PA) in ZFC, just as one can eventually >> find googolduplex digits of sqrt(2)+sqrt(3). So far, no poster or >> book has so convinced him. Therefore, there is no reason for >> him to believe that such a proof exists. > Of course. But it's just a mathematical theorem that virtually anyone > who is informed in the basics of the subject can do for himself. No > one can give a proof for Charlie-Boo that will convince him, since a > proof of this theorem depends on lots of terminology, formulations, > and previously proven theorems in Z set theory, which he refuses to > learn (let alone the predicate calculus). You don't expect someone to > prove in a post or two some result in mathematics such as analysis, > abstract algebra, etc. that requires first learning the basics of > those subjects, do you? Same for mathematical logic. I was wondering what position C-B would take on the model of Euclidean geometry provided by tuples of real numbers. The model dates back to Descartes (details filled in by various people up to Hilbert) and it's a bit more complicated to verify than PA in ZFC. But despite being a similar argument, it's in some sense more "ordinary" mathematics, so it doesn't get the cranks going. Or does it? Does C-B think algebraic geometry is illegitimate? ----------------------------------------------------------------------------- e m a i l : j a c k @ c a m p i n . m e . u k Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland mobile 07800 739 557 <http://www.campin.me.uk> Twitter: JackCampin
From: MoeBlee on 3 Jul 2010 17:26
On Jul 3, 1:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 1, 1:40 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> MoeBlee wrote: > >> A crank would "say" anything too! But I've never believed Aatu is a crank > >> so where's his _proof_, in FOL level or meta level? Oh, but you're going > >> to explain the "proof" right below, I see. > > > WHAT proof? Proof of WHAT? Proof of the consistency of PA? There are > > many. > > So there are many (formal) proofs of the consistency of PA, after all! OF COURSE! I've said that all along. And I've said that one is quite reasonable to view that such proofs have no epistemological value. This is all discussed beautifully in layman's terms in Franzen's book. OF COURSE if one doubts the consistency of PA, then a proof, from an even STRONGER theory, such as Z, provides no basis for alleviating said doubts. When I say there is a proof in this contexgt, I mean 'proof' in the technical sense of a formal proof, a derivation using recursive rules of inference with a recursive set of axioms, not necessarily in the sense of "indisputably convincing basis for belief" or related such senses. Such a thing may well not provide adequate basis for BELIEF if one does not already have adequate basis to believe said axioms are true. > > But Aatu is saying such proofs are NOT the basis for his > > conviction that PA is consistent. > > So are you saying Aatu is a crank, because he didn't base his assertion > that PA is consistent on proofs, but only on his "conviction", as you > claimed below? No, because my definition of the informal "sociological" rubric 'crank' (I've given it in other posts; I'm not going to type it up again) does not entail that one is a crank merely for having pre- formal mathematical beliefs. > >>> His basis is for that is not a FORMAL proof, but rather his > >>> conviction that the axioms of PA are true (and not even in confined to > >>> a FORMAL model theoretic sense of truth, but rather that the axioms > >>> are simply true about the natural numbers, as we (editorial 'we') > >>> understand the natural numbers even aside from any formalization. > >> Let me see: his proof > > >> - isn't based on rules of inference and axioms > >> - isn't based on "model theoretic sense of truth" > >> - is merely based on _conviction_ that "the axioms of PA are > >> true" and our intuitive knowledge of the natural numbers > >> "aside from any formalization". > > > I didn't say it is a PROOF. Why are you not LISTENING? > > Ah I see! So to you, Aatu just said things without proof here? Aatu and lots of people say things without proof. So what? Without proof I say that the lollipop I'm eating now is orange flavored. So what? Without proof, I say that I believe every theorem of PRA is true in a finitistic sense of 'true'. So what? > > I didn't post ANY definition of the consistency of a formal system. I > > posted a definition of consistency of a set of formulas. > > So, is the 2-formula set I gave a consistent set of formulas on your > definition? (The first one that you emphatically ended with "PERIOD.") I revised my definition (as I realized I had misstated my actual definition in my notes). If you want to know whether a certain set is consistent under that definition, refer to my revised definition. MoeBlee |