Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 27 Jul 2005 15:15 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Daryl McCullough said: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> >It seems like axioms are given a status that makes them >> >unquestionable and almost incomprehensible, beyond the application >> >of them. >> >> They are not incomprehensible to the people who work with them. > > Right. That's why nobody here seems to have recognized the basis for > the inductive axiom, and misses the fact that it constitutes an > infinite recursion. Sure. You guys really understand your tools. That's like saying one does not understand a lightbulb because one does not see that it is really a candle. The inductive axiom may be _motivated_ by unlimited recursion, but it does the same job without the effort. That's why it is a better choice of axiom than its motivation. And it really does not make sense to hold a match to the light bulb all the time. >> There purpose is to *clarify* what is going on, and they serve that >> purporse well. Axioms allow for precise communication between >> mathematical workers, and they allow for objective criteria for >> when a proof is valid or not. > > Axioms need to be proven too, from outside the system where they are > axioms, often inductively. Nope. They should have some motivation (just picking random stuff usually will lead to either self-contradictory or trivial systems); and when you are modeling some real-world phenomenon, you better make sure that your axioms make sense in that context. If they don't, this does not at all disturb the resulting mathematic truths, but you can't apply them to reality in the way you imagined. So axioms don't need to get proven. They don't even need to have real-world application or motivation, but if they don't, the results might, while mathematically true, be not useful in application. Not wrong, but useless. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Tony Orlow on 27 Jul 2005 15:18 Daryl McCullough said: > Tony Orlow (aeo6) wrote: > > > >Daryl McCullough said: > > >> But for the set we are talking about, there *is* no L. We're talking > >> about the set of *all* finite strings. That's an infinite union: If > >> A_n = the set of all strings of length n, then the set of all possible > >> finite strings is the set > >> > >> A = union of all A_n > >> = { s | for some natural number n, s is in A_n } > >> > >> This set has strings of all possible lengths. So there is no L > >> such that size(A) = S^L. > > >If those lengths cannot be infinite, then the set cannot be either. > > Why do you believe that? Read the proofs I provded today. > > >Either you have an upper bound or you do not, and if there is no > >upper bound on the values of the members, then they may be infinite. > > Why do you believe that? Because having no upper bound means that they can be any size, which includes infinite size. Otherwise, how do you knwo they are finite? > > >> You are assuming that every set of strings has a natural number L > >> such that every string has length L or less. That's false. > > > >I am saying that if L CANNOT be infinite > > I'm saying that there *is* no L. So don't talk about the case > where L is infinite or the case where L is finite. I'm talking > about the case where there *is* no maximum size L. Why do you chop the end of the statement? Made too much sense for you? If L CANNOT be infinite, if you CANNOT have infinite length strings, then you CANNOT have an infinite set of strings, UNLESS you have an infinite alphabet. The ONLY way to produce an infinite set of strings with a finite alphabet is to have infinite lengths for your strings, otherwise you have a finite, though "unbounded", set. > > Why do you think that there is a maximum size L? If there is no maximum size or finite upper bound, then L may be infinite, as I've said. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 15:21 Daryl McCullough said: > Tony Orlow says... > > > >I majored in Computer Science, and took plenty of logic and > >discrete math, and did quite well, thank you. > > I don't believe you. Did you take a course in mathematical logic? > Did you try telling your professors about your ideas about infinite > sets? yes, I gave them the answers they wanted, and told them they were wrong, which raised an eyebow, but I still got my A's. Jumping through the hoop doesn't mean you LIKE jumping through hoops. > > >Your inability to follow my argument is not an > >indication of my logical ineptitude. > > I didn't say that I can't *follow* your argument. I can follow > it just fine, in the sense that I can follow fallacious arguments > made by my children. No, you have repeatedly made statements that indicated you were getting lost. Respond to the two proofs I provided at your request, today. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Daryl McCullough on 27 Jul 2005 14:59 Tony Orlow (aeo6) wrote: >Daryl McCullough said: >> So, you agree that for *finite* sets, two sets have the same >> bigulosity if and only if there is a bijection between the two? >> But that no longer holds for infinite sets? >> >> Then how is bigulosity an improvement over cardinality? >Because Bigulosity takes into account the nature of the >bijection in order to determine a precise relative size of infinity. In other words, bigulosity is whatever you want it to be, and so you have a lot more flexibility. Just make it up as you go along. A = the set of natural numbers { 0, 1, 2, ... } B = the set of base ten numerals { "0", "1", "2", ... } C = the set of base two numerals { "0", "1", "10", "11", "100", ... } A and B have the same bigulosity. A and C have the same bigulosity. But B and C do *not* have the same bigulosity (clearly C has a smaller bigulosity than B). That's bigulosity for you... -- Daryl McCullough Ithaca, NY
From: Tony Orlow on 27 Jul 2005 15:23
Torkel Franzen said: > stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > > > >If those lengths cannot be infinite, then the set cannot be either. > > > > Why do you believe that? > > As we know, this idea is at the core of innumerable crank postings. > Experience strongly suggests that it is impractical to seek > elucidation of its roots by direct questioning of the authors of these > postings. Conceivably a decisive study could be made of it. No doubt > it has manifested itself through the ages, but the net archives > provide an unprecedently rich source of data about this peculiar > intellectual stumbling block. > How silly will you feel when you finally come to realize that this unending stream of "cranks" were all right, while you maligned, isnulted and bullied them? Will you ever apologize, or go to your graves in denial? Gee, I wonder.... -- Smiles, Tony |