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 17:00 Nam Nguyen wrote: > MoeBlee wrote: >> 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!) Argh! Too many typo's in a row: I meant "fwiw".
From: Nam Nguyen on 5 Jul 2010 18:02 Nam Nguyen wrote: > 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 fwiw, > I _think_ you know the answer but you don't want to say it!) <So to speak of course> So does this mean the capital "FOL" of the land of mathematical reasoning has been declared as "open" by the Platonic troops and the Godelian guards, and the forces of Relativity in reasoning could triumphantly march into the city's streets? Just wonder: what has happened to the Regular armies? They have been so silent for a long time now! (Or are they just planning for a fierce counter attack?) </So to speak of course>
From: Nam Nguyen on 5 Jul 2010 18:37 Aatu Koskensilta wrote: > MoeBlee <jazzmobe(a)hotmail.com> writes: > >> On Jul 5, 7:05 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>> "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: >>> >>>> Aatu said PA is consistent, _period_, without any formal proof? >>> There seems to be some confusion over my (perfectly standard as always) >>> take on these matters. >> Is there anything in my paraphrase several posts ago (in the full >> context I gave it) that is inaccurate? > > Not that I can see. It seems that it has given an incorrect impression > of what I think about these matters to Nam, though. I took your > paraphrase to be simply an explanation of what I mean when I say I don't > regard the usual consistency proofs for PA as establishing any > "relative" sort of consistency -- that is, I took you to be explaining I > were asserting > > PA does not prove P and not-P for any sentence P. > > and not merely > > ZF proves "PA does not prove P and not-P for any sentence P". > > or > > It is provable in ZF that PA does not prove P and not-P for any > sentence P. > > which is indeed what I intended. > >> However that quote is to be understood, for the record, I did not say >> that you hold there is no formal proof that PA is consistent, but >> rather that you hold PA is consistent on (for lack of better term I >> can think of right now) even more basic grounds than formal proof. > > Right. I hold PA is consistent because there's a mathematical proof of > this fact, that is, a piece of compelling mathematical reasoning > establishing the claim, invoking only principles we usually take for > granted when doing mathematics. In the vast ... vast realm of mathematical abstraction, there are always regions that "compelling mathematical reasoning" and "principles we usually take for when doing mathematics" would be of no value, because those are the regions where we're not doing our mundane ordinary mathematics! > I can't really see how anyone who didn't > take such things for granted could conclude from the "relative > consistency" theorems anything about the consistency of PA. I for one have never said one way or the other about the _actual_ (in)consistency of PA. I only said there's no effective way including relative consistency proof to determine that fact - whatever the fact be. > >> And that is what I have highlighted as to your view. > > It seems your posts have nevertheless led Nam and lwalke to incorrect > ideas about my views. This is not any fault of yours. I don't think I misunderstood your views at all here. Without MoeBlee's "representing" your views, my conversations, arguments with anybody on the subject would still be basically the same.
From: Nam Nguyen on 5 Jul 2010 19:28 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> MoeBlee <jazzmobe(a)hotmail.com> writes: >> >>> On Jul 5, 7:05 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>>> "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: >>>> >>>>> Aatu said PA is consistent, _period_, without any formal proof? >>>> There seems to be some confusion over my (perfectly standard as always) >>>> take on these matters. >>> Is there anything in my paraphrase several posts ago (in the full >>> context I gave it) that is inaccurate? >> >> Not that I can see. It seems that it has given an incorrect impression >> of what I think about these matters to Nam, though. I took your >> paraphrase to be simply an explanation of what I mean when I say I don't >> regard the usual consistency proofs for PA as establishing any >> "relative" sort of consistency -- that is, I took you to be explaining I >> were asserting >> >> PA does not prove P and not-P for any sentence P. >> >> and not merely >> >> ZF proves "PA does not prove P and not-P for any sentence P". >> >> or >> >> It is provable in ZF that PA does not prove P and not-P for any >> sentence P. >> >> which is indeed what I intended. >> >>> However that quote is to be understood, for the record, I did not say >>> that you hold there is no formal proof that PA is consistent, but >>> rather that you hold PA is consistent on (for lack of better term I >>> can think of right now) even more basic grounds than formal proof. >> >> Right. I hold PA is consistent because there's a mathematical proof of >> this fact, that is, a piece of compelling mathematical reasoning >> establishing the claim, invoking only principles we usually take for >> granted when doing mathematics. > > In the vast ... vast realm of mathematical abstraction, there are > always regions that "compelling mathematical reasoning" and "principles > we usually take for when doing mathematics" would be of no value, because > those are the regions where we're not doing our mundane ordinary > mathematics! > >> I can't really see how anyone who didn't >> take such things for granted could conclude from the "relative >> consistency" theorems anything about the consistency of PA. > > I for one have never said one way or the other about the _actual_ > (in)consistency of PA. I only said there's no effective way including > relative consistency proof to determine that fact - whatever the fact be. > >> >>> And that is what I have highlighted as to your view. >> >> It seems your posts have nevertheless led Nam and lwalke to incorrect >> ideas about my views. This is not any fault of yours. > > I don't think I misunderstood your views at all here. Without MoeBlee's > "representing" your views, my conversations, arguments with anybody on > the subject would still be basically the same. For argument sake, for example, we can assume Reflection Principle to say we "know" ZFC be consistent and from such "knowledge" we can "prove", not as a formal system theorem proof, that PA "is" consistent. What I'm saying is if we do that we may as well directly assume something like "The PA-Consistency Principle" which says something like there's a formula in L(PA) PA doesn't prove. I mean it'd be still as compelling to you _as well as to Nam_ just as the Reflection Principle would be compelling in ZFC case. But in the name of rigorousness of reasoning, how could we _assert_ PA is consistent, in the way we assert, say, Ax[~(Sx=0)] is provable in Q, or (x=x) /\ ~(x=x) is provable in T = {(x=x) /\ ~(x=x)}? And if we can't assert such thing in such way, _what are the compelling reasons_ that would prevent us from admitting so? I mean, if we don't actually know something isn't kind of prudent that we just say we don't know that something? (And I was blamed for being a crank?)
From: Nam Nguyen on 5 Jul 2010 19:46
Aatu Koskensilta wrote: > It seems your posts have nevertheless led Nam [...] to incorrect > ideas about my views. This is not any fault of yours. > Huh? You didn't say anything in that conversation and silently let him "represent" your views and if his "representing" goes wrong that would be my fault, no explanation needed? Where are your (and his) sense of credibility and responsibility? |