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 5 Jul 2010 14:47 MoeBlee wrote: > On Jul 5, 11:12 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 3, 3:07 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> MoeBlee wrote: >>>>> On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> MoeBlee wrote: > >> FOL syntactical proof is well defined, and _does NOT require_ the >> concept of "primitive arithmetic". > > I didn't say it does. > >>> And, in that regard, I have said all along that there is no finitistic >>> proof of the consistency of PA. >> So you've agreed that there's no formal proof for PA's consistency > > NO!!! ONE LAST TIME: > > There ARE formal proofs that PA is consistent. > > But, as far as I know, there is no FINITISTIC formal proof that PA is > consistent. > > As far as I can tell, formal proofs that PA is consistent carry no > basis for believing that PA is consistent if one already doubted that > PA is consistent. > >> and that >> if you go by formal proof only then you don't have knowledge of PA's >> consistency. > > I don't claim "knowledge" that PA is consistent. Rather, I find that > there is good basis (aside from formal proof) to believe that PA is > consistent. > > You did not at all understand my previous remarks about that. Oh I perfectly understand your point. It's the other way around. Usually in mathematical reasoning when we say such and such is provable in a formal system T _we'd assume by default T be consistent_ because otherwise why waste time talking about a theorem in an inconsistent theories? So let me rephrase my question in better clarity: Do you MoeBlee agree that there's no formal proof that PA is consistent in a consistent theory (formal system)? Very simple very straight forward question! > > At this point, I may elect (as is always the case anyway) to allow > your responses (including your plainly false statements (such as "So > you've agreed that there's no formal proof for PA's consistency") to > go without my remark. Life is just too short for trying to get through > to you. > > MoeBlee > >
From: Nam Nguyen on 5 Jul 2010 14:56 MoeBlee wrote: > On Jul 5, 11:36 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> The key question here is would the formal system containing "the formal >> proof that PA is consistent" be _itself_ consistent, >> according to Aatu? > > What Aatu claims about the consistency of the formal system doing the > proving is not "the key question" for me. > >> You seemed to know what his views be on this matter, can you answer >> this question? > > Yes. Aatu claims ZFC is consistent. He's written many posts on the > subject. Did Aatu "claim" or present _a formal proof_ that ZFC is consistent? > For even more on the subject, see Franzen's book Sorry. I always respect what TF had to offer. But I could not ask a dead man any question, which I'm sure I'd like to ask in the subject.
From: Nam Nguyen on 5 Jul 2010 16:06 MoeBlee wrote: > On Jul 5, 11:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 5, 11:12 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> MoeBlee wrote: >>>>> On Jul 3, 3:07 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> MoeBlee wrote: >>>>>>> On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>>>> MoeBlee wrote: > >> Do you MoeBlee agree that there's no formal proof that PA is >> consistent in a consistent theory (formal system)? > > (Everything I say here pertains to formal.) > > I do not know that there is no consistent theory that proves PA is > consistent. I take it that you meant to answer you don't know the answer to this question. (But you have to let me know if this is the case). (If I'm to answer the question then my answer is NO: there can be no such formal proof.) > And with a theory that has a sole non-logical axiom that is a > formalization of "PA is consistent", we do have a consistent theory > that proves PA is consistent. But, AGAIN, I don't draw any > epistemological import (basis for belief that PA is consistent) from > such a thing. > > And aside from such theories as just mentioned, I do have basis to > believe (without claiming certainty) that there exists a (non-trivial) > consistent theory that proves PA is consistent. However, I do not take > such a proof itself as a basis that one should believe PA is > consistent if one already had strong doubts that PA is consistent. > > And I've explained about all of that in previous posts (either in this > thread or in previous threads). Please read Franzen's book; as it > would save a lot of typing for both of us. I did read the portion of the book I have and from that I already concluded that TF actually didn't have an answer to the question I asked you above. If you believe he had done so, and I'm wrong here, you had to cite _a clear reference_ in his book where he'd answer the question somehow. A mere mentioning "Franzen's book" isn't sufficient to justify to an answering the question I asked.
From: Nam Nguyen on 5 Jul 2010 16:46 MoeBlee wrote: > On Jul 5, 1:06 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> I take it that you meant to answer you don't know the answer to this >> question. (But you have to let me know if this is the case). > > A question such as yours presupposes a framework from which the > question is asked. Agree. And usually when it's left unsaid it's assumed to be FOL= framework of formal axiom systems (which are syntactical). This is what I presupposed. (I thought we understood that when we talked - many times - about formal proofs, FOL, etc... No?) > > Without myself adopting whatever presuppositions you might or might > not have, I gave you a detailed and to the point answer to your > question from within my own modest framework. Now that you know what is my framework, I'd like to hear your straight answer so that we don't have to go back and forth. > > On the other hand, I'm not interested in a mode that is less an > explanation of certain notions and has more the character of a > deposition. > > > MoeBlee > > > >
From: Nam Nguyen on 5 Jul 2010 16:58
MoeBlee wrote: > On Jul 5, 1:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 5, 1:06 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> I take it that you meant to answer you don't know the answer to this >>>> question. (But you have to let me know if this is the case). >>> A question such as yours presupposes a framework from which the >>> question is asked. >> Agree. And usually when it's left unsaid it's assumed to be FOL= >> framework of formal axiom systems (which are syntactical). This >> is what I presupposed. (I thought we understood that when we >> talked - many times - about formal proofs, FOL, etc... No?) > > No, I mean even more general presuppositions (and, to be fair to you, > I don't expect you'd know that since I didn't specify what kind of > presuppositions I meant). Never mind though. It's not worth the ordeal > now of going into yet another issue as to what kind of presuppositions > I have in mind; I meant it merely in the sense of a GENERAL > disclaimer. > > ASIDE from that, I've failed in virtually every attempt to communicate > with you on virtually every matter, informal or informal, I've > discussed with you. I need to give up. You meant "informal or formal". You didn't fail on the "informal" part. It just on the "formal" part you failed, because you failed to give a clear cut answer. (And fwit, I _think_ you know the answer but you don't want to say it!) > > MoeBlee |