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 5 Jul 2010 14:47 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. For even more on the subject, see Franzen's book MoeBlee
From: MoeBlee on 5 Jul 2010 14:59 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. 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. MoeBlee
From: MoeBlee on 5 Jul 2010 15:03 On Jul 5, 11:56 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > 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? Aatu has answered this over and over already! > > 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. Would you please read the book? It's easy reading, not too technical. And instead of saying that there's questions he can't answer because he's dead, just read the book to see what questions he DOES address anyway. THEN perhaps you might have additional questions. MoeBlee
From: MoeBlee on 5 Jul 2010 16:16 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. 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. 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: MoeBlee on 5 Jul 2010 16:53
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. MoeBlee |