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 3 Jul 2010 17:39 MoeBlee wrote: >> So there are many (formal) proofs of the consistency of PA, after all! > > OF COURSE! I've said that all along. And I've said that one is quite > reasonable to view that such proofs have no epistemological value. > This is all discussed beautifully in layman's terms in Franzen's book. > OF COURSE if one doubts the consistency of PA, then a proof, from an > even STRONGER theory, such as Z, provides no basis for alleviating > said doubts. > > When I say there is a proof in this contexgt, I mean 'proof' in the > technical sense of a formal proof, a derivation using recursive rules > of inference with a recursive set of axioms, not necessarily in the > sense of "indisputably convincing basis for belief" or related such > senses. Such a thing may well not provide adequate basis for BELIEF if > one does not already have adequate basis to believe said axioms are > true. Let me try to summarize what you said above. There's no syntactical proof that can possibly confirm the fact that PA is consistent, if PA is in fact consistent. If this is what you meant I'm OK with that. And if that's the case, would you be able to make a stand and say we in fact just don't know PA if is consistent? That would help to save a lot of ridiculous arguments.
From: Nam Nguyen on 3 Jul 2010 17:42 MoeBlee wrote: > On Jul 3, 2:04 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >> >>> I didn't state that as MY view. I was telling you AATU's view. For >>> Christ sakes! >> I was asking Aatu, _not you_ to clarify about relative consistency >> and you "represented" him, even though he had never asked you to do >> so! > > I don't presume to represent him in general. Only as to those specific > remarks, I chose to state what his view is. At worst, I'm guilty of > being presumptuous to state what his view is. But I have sufficiently > discussed the subject of consistency of PA with Aatu and read enough > of his posts to have a clear understanding of his view to the extent I > stated it. Then don't attack people on the behalf of Aatu's views!
From: Nam Nguyen on 3 Jul 2010 18:07 MoeBlee wrote: > On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>>> So there are many (formal) proofs of the consistency of PA, after all! >>> OF COURSE! I've said that all along. And I've said that one is quite >>> reasonable to view that such proofs have no epistemological value. >>> This is all discussed beautifully in layman's terms in Franzen's book. >>> OF COURSE if one doubts the consistency of PA, then a proof, from an >>> even STRONGER theory, such as Z, provides no basis for alleviating >>> said doubts. >>> When I say there is a proof in this contexgt, I mean 'proof' in the >>> technical sense of a formal proof, a derivation using recursive rules >>> of inference with a recursive set of axioms, not necessarily in the >>> sense of "indisputably convincing basis for belief" or related such >>> senses. Such a thing may well not provide adequate basis for BELIEF if >>> one does not already have adequate basis to believe said axioms are >>> true. >> Let me try to summarize what you said above. There's no syntactical >> proof that can possibly confirm the fact that PA is consistent, >> if PA is in fact consistent. If this is what you meant I'm OK with >> that. > > By 'proof' in this context I mean formal proof. Totally agree. Like, Ax[~(Sx=0)] has a formal proof in PA, using strictly rules of inference, which is what is meant by formal proof! > > I don't know what you find bundled with the word 'confirm', so I > prefer to stand by what I posted, which is clear enough, especially as > it is an extremely common notion discussed widely. Oh very simple. I can _confirm_ by sheer formal proof that the theory T = {(x=x) /\ ~(x=x)} is inconsistent. > >> And if that's the case, would you be able to make a stand and say we >> in fact just don't know PA if is consistent? > > I don't view the matter in that simple way of putting it even. So the question is, MoeBlee, can you confirm PA is consistent, the way I can confirm beyond any question this T is inconsistent? That's very simple, but genuine, technical question I'd think a beginner of Mathematical Logic would understand. And at the risk of being wrong, I'll venture out and state that MoeBlee can't confirm that (and I have reasons for that). What about you MoeBlee: can you confirm that PA is actually consistent? > >> That would help to save a lot of ridiculous arguments. > > You'd do best not to argue as if I hold certain views I've never > stated holding. Just a plain question for you MoeBlee. Are you going to be able to answer that question? >
From: Nam Nguyen on 3 Jul 2010 18:12 MoeBlee wrote: > On Jul 3, 2:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Then don't attack people on the behalf of Aatu's views > > I didn't! > > You're hopeless. You said to me: > I'm sorry, Nam, [...] reading your > posts with others, I just have to conclude that communication on these > subjects can't be achieved with you. Sorry, MoeBlee, you were wrong (if not hopeless) about attacking people, when representing a portion of somebody else's views! (Note the your word "these subjects"!) Don't blame it on me!
From: Nam Nguyen on 4 Jul 2010 02:21
MoeBlee wrote: > It's likely this post will earn me yet more tedium in responding back > and forth with nutcases, but I'd like to add this anyway: > > Consider the axioms of PA aside from the induction schema. My view is > that IF there are ANY non-logical simple truths about natural numbers > (in some suitable pre-formal ordinary sense of 'natural number', which > would be any suitable abstraction of the activity of counting) then > those axioms are among them. That is to say, if there is ANYTHING that > could be said to be a finitistic mathematical truth, then those axioms > are among them. Close to 300 years ago it seems to me mathematicians such as Goldbach and Euler also had quite similar "ordinary sense of 'natural number'" when they investigated the "truths about natural numbers" viz a viz Goldbach Conjecture, which I believe to them the naturals could be expressed in a few axioms not that different from our modern day non-induction ones of PA. The point is whatever the historical reason we need them (e.g."counting") the concept of the naturals has come to us with certain baggage: if we take the multiplication and/or the prime numbers out, the naturals are no longer what we'd envision them to be. But if we leave them in, certain unknowable would be, in your own wording below, _ineluctable _. And these are intrinsic unknowable whether or not we mention the word "induction" (since the naturals are supposed to also reflect truths about Q's theorems). The point is that PA's axioms, together, would present an incomplete picture about what we think of as the naturals, and that if we accept the naturals we have to accept both the good and the bad of such concept. We can't filter out only the good part (the induction one) and call it the natural numbers, because if we do so we'd have to eliminate either multiplication or prime numbers and what we'll have left is a (negation) complete but degraded PA system! > > Then, the induction schema seems to me ineluctable too (though, not as > "certain" as the finitistic part of PA, or at least not certain in the > same way, as there are complications having do with the notion of > properties, even confined to those properties that are "captured" by > first order formulas, etc.) . But as alluded above, the issue of understanding the naturals or the purported consistency of PA already starts with Q. PA at best doesn't change the issue and at worst would complicate it. > > Now, I do not hold to this perspective as a matter of dogma, nor do I > have any polemical investment in it. Rather, this perspective is just > that: a perspective for me. It is a basis for "making sense" of > mathematics for me. If someone else rejects it as a basis, then I'm > not inclined to try to convince otherwise. In other words, such a > perspective is for me a "way of looking at things" that is suitable, > that serves me well, at least so far in my life. I don't need to argue > that it is a "correct" perspective, let alone a "true" perspective. > Rather, it is, for me, just a perspective that serves me well toward > making sense of mathematics and of my own mathematical mental > experiences. Of course one would have one's own perspective on certain issues in life, including perspective in mathematical reasoning, and one could choose a different logic. But if one accepts a reasoning framework, FOL e.g., with definitions of inference rules, proof, model, or what have we, then one should adhere to them and not to introduce _individual_ "convictions", intuitions that are not only outside what's permissible by the framework but also inconsistent or even detrimental to using the framework. Otherwise, it's just difficult for individuals to communicate and better the knowledge of sciences that mathematics is supposed to be the language of. > > So, in this way, certainty is put in terms that seems as unassailable > as could be to me: IF there are any certain (non-logical) mathematical > truths, then those of finitistic arithmetic are ones. I can't imagine > that they are not certain, but even IF I could imagine them not > certain, then I don't know what OTHER (non-logical) mathematical > certain truths there can be. But why should mathematical truths be certain (i.e. absolute)? We used to hold a lot of "truths" as certain too: the 5th postulate, 1+1=2, the simultaneity of certain 2 events, the clock-work universe, .... They all turn out to be relativistic, right? Besides, if you hold certain formulas in L(PA) for truths, why not, say cGC? Of course Aatu would suggest each of us is free to hold true what each so chooses. But isn't mathematics supposed to be communicable universally? > And even here, I don't disallow that > there might be even more certain non-logical truths than those of > finitistic arithmetic. It's just that presently I can't imagine them. > I can't imagine what non-logical matter would qualify as certain if > simple matching of strings of symbols is not certain. I probably miss something here but what does "matching of strings of symbols" have to do with formula truth or with certainty of formula truth? Could you elaborate? > > However, if someone doesn't even regard finitistic mathematics (such > as results of PRA) - essentially just recognizing whether strings of > symbols match or do not match - as "certain", "correct", whatever, > then I admit that I can't see what basis for communication I would > have with him or her. I don't know how we could even communicate if we > couldn't agree that we can look at finite strings of symbols and check > for matching. You've lost me here I'm afraid. Is PRA a formal system? If so, why can't you and the person communicate with each other the results of formal proofs using rules of inference. It's a curiosity that we've been talking about formal systems such as PA and yet in this not-too-short post you've not mentioned inferences by FOL rules of inference. Would there be reasons why you've not mentioned them in this post? |