Obections to Cantor's Theory (Wikipedia article)
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From: Chris Menzel on 20 Jul 2005 12:45 On Wed, 20 Jul 2005 09:08:02 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > ... >> To say that a list of the reals is aleph_1 in length assumes the >> continuum hypothesis; is that what you intend? (And if so, why?) >> Moreover, granted, aleph_1 is omega_1 in pure set theory, but one should >> use ordinal numbers rather than cardinals when talking about such things >> as the length of a list, as there are many lists of different ordinal >> lengths that are the same size. > I don't think it assumes any such thing. If the assumption you are referring to is that there is a list of reals of length omega_1 (= aleph_1 in pure set theory), then all you are doing by denying that this assumption requires the continuum hypothesis is showing your abject ignorance of the subject. > In fact, it is my position that there is an infinite spectrum of > infinities between that of the naturals and that of the reals. Well, then your assumption that there is a list of reals of length omega_1 is curious indeed. > If set theory considers the set of reals to have cardinality aleph_ 1, > which is strictly larger than the number of naturals, aleph_0, and you > have a list of reals each with a number of digits equal to the number > of naturals, then you do not have a diagonally traversible square, but > a rectangular grid of digits. Er, uh, well, wait a minute. I've really gotten confused about what you're doing now. You seem to be starting your representation of Cantor's Theorem with the conclusion of Cantor's Theorem -- that there are more reals than natural numbers. Of course there is no "diagonally traversal" square in that case. But Cantor's Theorem starts with the assumption that there is a list of length omega. >> Pretty clearly, you aren't terribly well-educated in set theory. >> Don't you think you should understand a field before you try to point >> out its flaws? And don't you see that it makes you look rather >> silly? Would you consider attacking superstring theory without >> understanding basic quantum mechanics? > > I have not immersed myself deeply in set theory, no. But, I can see > clearly that some of the conclusions are wrong, and in arguing this, I > have stumbled upon a few of the obvious flaws in the logic. David Kastrup has noted the problems with this reasoning. Many thinkers through the years have thought they clearly knew one thing or another, only to be shown wrong by deeper advances in the relevant discipline. As I noted earlier, actually learning the material you criticize as an intelligent but uneducated amateur will help you to see where your conceptual errors lie. > Hopefully I will find time soon to decipher the specific axioms of ZF > and see whether the roots of the problem lie there, or with subsequent > assumptions. Or with your own thinking -- an important part of being a scholar is always to bear in mind that one *might* be wrong. > Now, if I came up with a theory that "proved" that 1=2, would you > believe it, at first glance? Of course not at first glance. But I would not dismiss it out of hand, either, given what we know from Gýdel about the unprovability of consistency. I would be greatly skeptical, of course, if the proof came from someone who obviously doesn't know much about arithmetic or set theory. > Should I accept a theory that "proves" that we can take one ball, cut > it into a finite jumber of pieces, and reassemble it into two solid > balls, each the same size as the original? The Banach-Tarski "paradox" > was offered as a disproof by contradiction of the axiom of choice, Actually, it wasn't. > although I am not sure that axiom is directly at fault for what is > obviously an incorrect result. I'm not sure why not, as that axiom is the key to deriving the "paradox" in question, and the other axioms involved are quite innocuous by comparison. > Somehow, instead of being considered a proof of an inconsistency in > set theory, this result has been accepted as a counter-intuitive > "fact", which I simply cannot accept as having anything to do with > reality. Yes, I am one of those that believes that real math reflects > some actual reality, whether we have discovered the application yet or > not. Lots of people agree. The interesting fact, however, is that there is no *formal* inconsistency in the B-T theorem. A genuinely mathematical response, then, is to try to understand why, which might lead to an investigation and modification of the axiom of choice; or to some modifications in topology; or perhaps to a weaker set theory like KPU from which it can't be derived. That's what a real mathematician would do. >> > and the antidiagonal simply exists on the list, below the line of >> > diagonal traversal. >> >> Aside from your tenuous grasp of basic transfinite arithmetic, the >> critical terms in your argument -- "digital number", "length", >> "exponentially longer" (cardinal or ordinal exponentiation?), >> "width", "diagonal traversal", "below the line of diagonal traversal" >> -- are much too vague for your argument to be evaluated. For all we >> know, you might have some genuine insights. But currently, your >> argument is smoke and mirrors; it hasn't been expressed as >> mathematics. > > I think you know exactly what i mean by each of those terms. Actually, I haven't a clue. You've already shown that you don't understand a number of very elementary notions in the subject you purport to criticize. This gives no confidence that you understand to ones in question. > They are all widely understood. It is typical for Cantorians to resort > to claims of vagueness on the part of their opponents, while > presenting such vague proofs as the diagonal argument, Hang on, buddy, lies and innuendo will get you nowhere. As noted above, there is a completely rigorous, demonstrably valid proof of Cantor's Theorem in ZF that all "Cantorians" know. But look, I'm more than happy to play fair -- do some real mathematics: Define *each* of the terms above, and lay our your argument step by step like a real mathematician. Chris Menzel
From: Alan Morgan on 20 Jul 2005 13:03 In article <42de4a98$9$fuzhry+tra$mr2ice(a)news.patriot.net>, Shmuel (Seymour J.) Metz <spamtrap(a)library.lspace.org.invalid> wrote: >In <dbjqog$tg7$1(a)xenon.Stanford.EDU>, on 07/19/2005 > at 09:19 PM, amorgan(a)xenon.Stanford.EDU (Alan Morgan) said: > >>No, you didn't. You proved this for finite sets only. You claimed, >>without proof, that this result applied to infinite sets. > >The result does apply to infinite sets. There is no largest natural in >an infinite set of naturals, and hence any statement about the size of >"the largest natural" is vacuously true. Unfortunately, tony doesn't >understand that proving flying pigs are reptiles is not useful when >there are no flying pigs. Good point. Probably too subtle for Tony to understand, but still a good point. Alan -- Defendit numerus
From: Daryl McCullough on 20 Jul 2005 12:54 Tony Orlow writes: > >Dik T. Winter said: >> Back on your horse again. Tell me about the binary numbers (extended to the >> left with 0's) where the leftmost 1 is in a finite position. Are all those >> numbers finite? Are there only finitely many of them? > >yes and yes What definition of "finite" are you using? Once again, if your claims had any merit whatsoever, then you would be able to rephrase them in a way that does not rely on unorthodox meanings of terms. Rephrase your claim without using the word "finite" or "infinite". Is that possible? -- Daryl McCullough Ithaca, NY
From: malbrain on 20 Jul 2005 13:17 Tony Orlow (aeo6) wrote: > David Kastrup said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > David Kastrup said: > > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > >> > > >> > David Kastrup said: > > >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > >> >> > > >> >> > Alec McKenzie said: > > >> >> >> "Stephen J. Herschkorn" <sjherschko(a)netscape.net> wrote: > > >> >> >> > > >> >> >> > Can anti-Cantorians identify correctly a flaw in the proof > > >> >> >> > that there exists no enumeration of the subsets of the > > >> >> >> > natural numbers? > > >> >> >> > > >> >> >> In my view the answer to that question a definite "No, they > > >> >> >> can't". > > >> >> >> > > >> >> >> However, the fact that no flaw has yet been correctly > > >> >> >> identified does not lead to a certainty that such a flaw > > >> >> >> cannot exist. Yet that is just what pro-Cantorians appear to > > >> >> >> be asserting, with no justification that I can see. > > >> >> >> > > >> >> > Even though every subset of the natural numbers can be > > >> >> > represented by a binary number where the first bit denotes > > >> >> > membership of the first element, the second bit denotes > > >> >> > membership of the second element, etc? > > >> >> > > >> >> Well, what number will then represent the set of numbers dividable by > > >> >> three? > > >> > 100100...100100100100 > > >> > > >> > Of course, you will argue that this infinite value is not a > > >> > natural number, since all naturals are finite, but that is > > >> > clearly incorrect, as it is impossible to have an infinite set of > > >> > values all differing by a constant finite amount from their > > >> > neighbors, and not have an overall infinite difference between > > >> > some pair of them, indicating that at least one of them is > > >> > infinite. > > >> > > >> You have not shown such a thing, and of course it would be > > >> inconsistent with the Peano axioms defining the naturals. > > > > > > That is simply not true. > > > > Sulking won't help. > Don't be a jerk. > > > > > There is nothing in Peano's axioms that states explicitly that all > > > natural numbers are finite. > > > > It is an immediate consequence. > Read on. > > > > > The fifth axiom, defining inductive proof, is used to prove this > > > theorem, but it is a misapplication of the method. > > > > An axiom is not a "misapplication". > No, the proof is a misapplication of the axiom. Yes, you're right, the axiom is a statement of agreement. We agree via the axiom that the definition of ALL is EACH-AND-EVERY. Each and every natural number is finite. That's our concept of the natural numbers. We agree that we need only deal with a finite number of steps to prove something about the infinite set of natural numbers. (...) > Now, I have heard the argument that inductive proof does not prove things for > an infinite number of steps, but only a finite number, but if this is the case, > then it does not prove anything for an entire infinite set of natural numbers. That's not what our axiom says. We get to prove things about the infinite set of natural numbers in a finite number of steps. Applying the axiom of induction is a single step in a leap of faith that we agree to before hand. > Either you agree that there are an infinite number of steps involved, or that > the set of naturals is finite, or that inductive proof does not prove a > property true for all natural numbers as Peano stated. That's not what our axiom says. It says that induction covers all the natural numbers in a single step, a single leap-of-faith. karl m
From: Virgil on 20 Jul 2005 13:22
In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Chris Menzel said: > > Pretty clearly, you aren't terribly well-educated in set theory. Don't > > you think you should understand a field before you try to point out its > > flaws? And don't you see that it makes you look rather silly? Would > > you consider attacking superstring theory without understanding basic > > quantum mechanics? > I have not immersed myself deeply in set theory, no. But, I can see clearly > that some of the conclusions are wrong Without knowing how and why those conclusions have been reached? What sort of crystal ball are you using? |