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 3 Jul 2010 01:55 Nam Nguyen wrote: > Transfer Principle 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? 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? > > I agree with you in the above: there's a degree of being double > standard that Moeblee and other "standard theorists" have exercised: > they'd blast people as talking nonsense if people don't conform to > standard logics 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_! > > The most interesting question is why MoeBlee and those "standard theorists" > never admit they're just being philosophical about PA's consistency, > 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". > > Then why did the founders of FOL invent FORMAL proofs in the first > place? So that MoeBlee and others in that group could throw these formal > proofs away like boring toys, whenever they feel they'd like to? > > If that's isn't being double standard in arguing, reasoning then I don't > know what is. For the record, I've never stated I don't believe in the in the consistency of PA. I actually don't know how to believe one way or another. My position though is that if PA is genuinely consistent we simply can't know such fact! So, it's _an invalid reasoning to assert_ anything on the account of such unknowable (again if PA is consistent). In practice, for the typical mathematics that doesn't have to express any kind of relativity, such as SR, we could close our eyes to such unknowable and assume we "know" PA is consistent. But to adequately express SR, for example, we can't ignore such unknowable. For what it's worth, Godel came very close to being an Einstein in realm of mathematical abstraction. Not quite so nonetheless.
From: Nam Nguyen on 3 Jul 2010 11:05 R. Srinivasan wrote: > 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. This is where the "alliance" between NAFL forces and the relativists, so to speak, would end. An inconsistency of a theory should be formalizable, in the sense there there would be a (finite) syntactical proof for it!
From: Nam Nguyen on 3 Jul 2010 11:50 R. Srinivasan wrote: > On Jul 3, 8:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> R. Srinivasan wrote: >>> 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. >> This is where the "alliance" between NAFL forces and the relativists, >> so to speak, would end. An inconsistency of a theory should be formalizable, >> in the sense there there would be a (finite) syntactical proof for it! > >> > Sure. Such a proof of a contradiction P&~P would indeed be > formalizable in an appropriate NAFL theory T. We can indeed conclude > from this formal proof that "T is inconsistent". However, the > proposition "T is inconsistent" is not formalizable in the language of > T. > The notion of provability is not formalizable in T and hence "T |- > (P&~P)" cannot be a proposition of T. Let P <-> x=x. If T is inconsistent, the formula (P&~P) is such a formula in [_any_] L(T).
From: Nam Nguyen on 3 Jul 2010 16:37 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! > 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? > >>> 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? > 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.")
From: Nam Nguyen on 3 Jul 2010 17:04
MoeBlee wrote: > > I didn't state that as MY view. I was telling you AATU's view. For > Christ sakes! I was asking Aatu, _not you_ to clarify about relative consistency and you "represented" him, even though he had never asked you to do so! So you did represented Aatu in his absence! Therefore in this context I can't make a distinction between you or him. If you don't want it that way, then don't represent anybody! Just argue on your own. As far as I'm concerned, you just hide behind Aatu's name to "blast" me (and you did here a few times) without using your own credibility! |