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 26 Jun 2010 10:42 R. Srinivasan wrote: > On Jun 26, 1:58 am, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > [...] >> You may wish to know that ZFC with the axiom of infinity replaced by its >> negation is a model of PA and vice versa. >> >> > There are two notions of consistency, namely the syntactic and model- > theoretic notions, which are supposed to be equivalent. > Syntactically the consistency of PA is expressed by the sentence > Con(PA) which can be encoded in ZF and proven. What does it mean for a formula A of L(T) to _syntactically signify_ the (possible) consistency of T?
From: Nam Nguyen on 26 Jun 2010 17:42 R. Srinivasan wrote: > On Jun 26, 7:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> R. Srinivasan wrote: >>> There are two notions of consistency, namely the syntactic and model- >>> theoretic notions, which are supposed to be equivalent. >>> Syntactically the consistency of PA is expressed by the sentence >>> Con(PA) which can be encoded in ZF and proven. >> What does it mean for a formula A of L(T) to _syntactically signify_ the >> (possible) consistency of T? >> > By the way I do not agree that a formula of L(T) can express the > consistency of T. So why did you use the phrases "syntactically" and "consistency of PA" in your "Syntactically the consistency of PA is expressed by the sentence Con(PA)"? > As I have stated in my post later on, I strongly > believe that the consistency of T is a metamathematical (or in this > case metatheoretical) notion that cannot be expressed in the language > of T. So, again, why did you say "the consistency of PA is expressed by ... Con(PA)", as below? >>> Syntactically the consistency of PA is expressed by the sentence >>> Con(PA) which can be encoded in ZF and proven. > > However, according the conventional wisdom, which is what I was > stating above, a formula A of L(T) can represent the (code of the) > assertion that "There does not exist a proof of '0=1' in the theory > T", for theories T that can encode a certain amount of arithmetic. At > least this is what Godel claimed. But all this is still syntactical, which you said above that "I do not agree that a formula of L(T) can express the consistency of T". No?
From: Nam Nguyen on 27 Jun 2010 12:14 R. Srinivasan wrote: > On Jun 27, 2:42 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> So why did you use the phrases "syntactically" and "consistency of PA" >> in your "Syntactically the consistency of PA is expressed by the sentence >> Con(PA)"? >> >> > As I said, I was expressing the conventional wisdom when I said that. > I was arguing from the point of view of accepted classical logic. > Basically I was playing along with the status quo to make a point > later on. >> >>> As I have stated in my post later on, I strongly >>> believe that the consistency of T is a metamathematical (or in this >>> case metatheoretical) notion that cannot be expressed in the language >>> of T. >> So, again, why did you say "the consistency of PA is expressed by ... >> Con(PA)", as below? >> > That is an accepted result of Godel from classical logic. Later on, > when I was talking about NAFL, I disagreed with the classical result. > NAFL is what I really believe in. I see your points now. Thanks for the clarification. For what it's worth, it's going to difficult to persuade some "standard theorists" on the value on some classical results: they do have their own _beliefs_ in their mind and - I might say - heart!
From: Nam Nguyen on 29 Jun 2010 23:47 Aatu Koskensilta wrote: > Frederick Williams <frederick.williams2(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? (Iow, you didn't mean it's a fact that PA is syntactically consistent, did you?) > The interest in Gentzen's > proof lies elsewhere. >
From: Nam Nguyen on 30 Jun 2010 00:09
R. Srinivasan wrote: > On Jun 30, 1:36 am, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Jun 29, 12:28 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote >> >> to much flotsam still for me to spend more time than I've already >> spent. >> >> However: >> >>>> We PROVE from ZF-Inf that there IS NO SUCH object that you are calling >>> '> D'. (or at least we have not before us a proof that there IS such an >>>> object). Just adding a constant symbol 'D' and saying whategver you >>>> want about it does not override. >>> You do not have any such proof. >> I SAID, "or at least we have not before us a proof that there IS such >> an object". >> >> But it's simple anyway: >> >> Therorem of ZF-I: >> >> Ex~En x in P_n(0) -> ~EyAz(zey <-> ~En z in P_n(0)) >> >> Proof: Toward a contradiction suppose Ex~En x in P_n(0) and >> Az(zey <-> ~En x in P_n(0)). >> Let ~En x in P_n(0). >> Let j be arbitrary. >> ~En xu{j} in P_n(0). >> So Aj j in Uy. >> >> Theorem of ZF-I: >> >> ~Ex~En x in P_n(0) -> Ey(Az(zey <-> ~En z in P_n(0)) & y=0) >> >> Proof: Immediate. >> >> Then, as far as I know (which is pretty limited) it is not decided in >> ZF-I whether Ex~En x in P_n(0). Someone may inform me further on that, >> but I'm pretty sure that ZF-I doesn't tell us whether there are or are >> not sets other than the hereditarily finite sets. >> > I think it is not known whether this proposition (That there are sets > other than the hereditarily finite sets) is undecidable, refutable, or > provable in ZF-I. Undecidability of this propostion is just an > assumption as far as I know. >>> How can something be true >>> "in the standard model of PA iff PA is inconsistent" ????? >> Typo of omission. I meant, "true in the standard model for the >> LANGUAGE of PA", as I had posted in previous messages. >> > Hmmm. I am not very conversant with classical model theory. So > according to you, there is a "standard model for the LANGUAGE of PA" > even if the theory PA is inconsistent. MoeBlee has to clarify but I'd think that's what he meant. > May I infer that you have used > infinite sets to define this model? How can you do that if the theory > PA is inconsistent (which would make ZFC inconsistent as well)? 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! > > Anyway, in NAFL there is no such thing as the "standard model for the > LANGUAGE of PA". > Truths are with respect to (consistent) axiomatic > theories and there are no truths in just the language of a theory. I don't know much about NAFL but I'd agree with 1st half of this statement. |