Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 26 Jul 2005 18:17 In article <MPG.1d506cc1bf3b4b0c989fa0(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > imaginatorium(a)despammed.com said: > > Tony Orlow (aeo6) wrote: > > > > > I don't see where you pointed out any specific flaw, except to rant about > > > your > > > largest finite number again. > > > > No, well, I give up. Just for my curiosity, though, I still cannot > > understand your point when you complain about "ranting about my[sic] > > largest finite number". It has been pointed out to you so many times - > > with absolutely no effect - that the Peano axioms (or any similar more > > informal notion of pofnats) imply that there cannot be a largest > > pofnat. Just tell me: do you claim... > > > (sigh) > > (1) There _is_ a largest pofnat. > no > > (2) There is no largest pofnat (but the contradictions with your ideas > > escape you) > yes, please explain the contradiction, without the mantra. I have heard > Virgil > claim that I think there is one, or that I MUST produce one, if I am to claim > there are infinite whole numbers. I see no such need. I ahve agreed that one > cannot count finitely from the finite to the infinite, and it has been agreed > that one cannot count down from the infinite to the finite. The first fact > does > not mean the infinite whole don't exist, any more than the second means that > finite wholes cannot exist. So, where is the contradiction? If one cannot count from finite to infinite, Peano's fifth axiom eliminates any possibility of any infinite naturals. So that TO must either give up infinite naturals or give up working with any system incorprating the Peano axioms. He can't have both.
From: Daryl McCullough on 26 Jul 2005 18:11 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? >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? >> 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 think that there is a maximum size L? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 Jul 2005 18:17 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? >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. -- Daryl McCullough Ithaca, NY
From: Torkel Franzen on 26 Jul 2005 18:41 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.
From: malbrain on 26 Jul 2005 18:42
Randy Poe wrote: > malbrain(a)yahoo.com wrote: > > David Kastrup wrote: > > > Of course not. But the basis for choosing an axiom is irrelevant to > > > the application of the axiom. And this axiom was chosen exactly in a > > > manner that does not require recursive application. > > > > When working a proof one chooses one's axioms based on the desired > > result. How can you say that the choice is irrelevant to the > > application? > > Eh? > > Can you give an example that you think illustrates this > "choosing one's axioms based on the desired result"? Do > you think different number theories decide which of the > Peano axioms to accept and which not, or add additional > axioms of their own, based on a "desired result"? What is the axiom of infinity if not an addition to the Peano axioms? Didn't the axiom of induction come first? karl m |