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 19:51 Nam Nguyen wrote: > 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? Hopefully by now you'd understand why in the other thread I asked why in debates you've tended to: > think you're infallible and above the rigorousness of mathematical > reasoning?
From: Nam Nguyen on 7 Jul 2010 22:05 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> MoeBlee wrote: >> >>> 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 and that >> if you go by formal proof only then you don't have knowledge of PA's >> consistency. > > There are many formal proofs of the consistency of PA. None of them are > finitistic. So it doesn't seem you used the phrase "formal proof" in the standard way that textbook (e.g. Shoenfield's) would use. In that standard usage, a formal proof is a (finite) syntactical proof of a FOL formal system theorem. Given that standard definition of "formal proof", would you agree with my statement above that: >> there's no formal proof for PA's consistency and that if you go by >> formal proof only then you don't have knowledge of PA's consistency. ? -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: Nam Nguyen on 7 Jul 2010 22:19 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> 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?) > > What we take to be known in mathematics is to some extent a matter of > opinion. I don't disagree with that actually, in a high level. But _then_ that means some of what we know in mathematics is _relative, subjective_ right? > In ordinary mathematics it's a triviality that PA is provably > consistent. So PA-consistency's being true is only of a matter of opinion, as you've alluded above, and not a matter of _fact_ such as it's a fact that PA formally proves Ax[~(Sx=0)], right? > Those who regard the principles and modes of reasoning taken > for granted in ordinary mathematics as dubious will of course have to > inspect the proof for themselves and make up their own mind. For what it's worth, I think before one cares about such "formal proof", one should inspect the _actual standard formal proofs_ of some theorems to see if PA would be inconsistent in the first place. -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: Nam Nguyen on 7 Jul 2010 22:22 Marshall wrote: > On Jul 7, 7:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>> MoeBlee wrote: >>>>> 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 and that >>>> if you go by formal proof only then you don't have knowledge of PA's >>>> consistency. >>> There are many formal proofs of the consistency of PA. None of them are >>> finitistic. >> So it doesn't seem you used the phrase "formal proof" in the standard >> way that textbook (e.g. Shoenfield's) would use. In that standard >> usage, a formal proof is a (finite) syntactical proof of a FOL formal >> system theorem. >> >> Given that standard definition of "formal proof", would you agree with >> my statement above that: >> >> >> there's no formal proof for PA's consistency and that if you go by >> >> formal proof only then you don't have knowledge of PA's consistency. >> >> ? >> > > Give it up, loon. PA is provably consistent. Deal with it. That's all the _technical_ arguments an intellectual clown like you, Marshall, could ever say! -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: Nam Nguyen on 7 Jul 2010 22:24
Marshall wrote: > On Jul 7, 7:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>> MoeBlee wrote: >>>>> 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 and that >>>> if you go by formal proof only then you don't have knowledge of PA's >>>> consistency. >>> There are many formal proofs of the consistency of PA. None of them are >>> finitistic. >> So it doesn't seem you used the phrase "formal proof" in the standard >> way that textbook (e.g. Shoenfield's) would use. In that standard >> usage, a formal proof is a (finite) syntactical proof of a FOL formal >> system theorem. >> >> Given that standard definition of "formal proof", would you agree with >> my statement above that: >> >> >> there's no formal proof for PA's consistency and that if you go by >> >> formal proof only then you don't have knowledge of PA's consistency. >> >> ? > > Give it up, loon. PA is provably consistent. Deal with it. That's all the _technical_ arguments an intellectual clown like you, Marshall, could ever say! -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ---------------------------------------------------- |