Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 27 Jul 2005 23:41 In article <MPG.1d51a81cd5e42d8a989fbf(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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. They are no less corrupt than any of their predecessors. > > > > >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? By finding an injection from the set of them to a proper subset, of course, how else? > > > > >> 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. Since all such "unbounded" ordered sets of finite elements have injections into proper subsets, they are Cantor-infinite, which is the only infinite of set size that counts here.
From: Virgil on 27 Jul 2005 23:44 In article <MPG.1d51a8af118aaa0d989fc0(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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. Since what TO provides never qualifies as a proof, either mathematically, or by his own confession above, logically, no response is needed.
From: Virgil on 27 Jul 2005 23:46 In article <MPG.1d51a923cd149b5b989fc1(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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.... Since this will take a good deal longer to occur that the instanciation of any of TO's infinite naturals, don't anybody except TO hold your breath.
From: imaginatorium on 27 Jul 2005 23:46 Tony Orlow (aeo6) wrote: > imaginatorium(a)despammed.com said: > > Tony Orlow (aeo6) 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? > > > > You claim, simultaneously that: > > > > (a) There is no largest pofnat. > > (b) There are only a finite number of pofnats. Please answer: have I misunderstood, or misrepresented your claims? (If so, which one?) Of course, I think that (a) is true, and (b) is false, but I am discussing *your* argument. OK? > > You suddenly became very explicit (elsewhere in the thread) and appear > > to agree that to say a set is *finite* means that it can be counted > > against a ditty, and the ditty stops. Is this right? You accept the 'ditty' definition of finite? If I start pointing at the elements one by one, reciting the ditty, then if the set of elements is finite, at some point in the (real, finite) future, the ditty will stop because the elements are exhausted. Yes? You agree? > > OK, so arrange the pofnats in normal ascending order, and count them > > against a ditty. In (b) above, *you* assert that the pofnats form a finite set. If you accept the 'ditty' definition of a finite set, this means that *you* must deduce from *your* assertion, that: > > When the ditty stops, you have reached the last > > pofnat, which is therefore the largest. This is a consequence of (b), > > and contradicts (a). > How about you tell ME where the ditty stops, and the values for the naturals > end, if you're so sure they are all finite? As soon as you end your ditty on > the element values, my ditty on the set size will be complete. Deal? I do not think the ditty stops, because I think the set of pofnats is infinite. My ditty "on the element values" (what other sort of ditty is there?) does not, in my opinion, stop. You appear to be claiming that it does. You also claim this does not contradict anything. Hmm. > > > > > > > > > > (3) The answer to "Is there a largest pofnat?" is somehow neither 'Yes' > > > > nor 'No'. > > > No, the answer is no, just like the answer to "is there a smallest infinite > > > number?" There is no distinct line between the finite and infinite. That line > > > is infinitely wide, and requires an infinite difference to cross. > > > > I wouldn't call it a "line", personally, but roughly speaking this > > appears to be a description of the set of surreals {0, 1, 2, 3, ...} U > > {..., w-3, w-2, w-1, w} > > (using union notation to prevent dotty confusion). > It is similar, if not the same. I started reading about the surreals, and it > seemed liek avery sensible approach, though I have not had time to read it all. > Hopefully during vacation. Unless scales fall from your eyes, I fear you will get nowhere with it. But worth trying. Brian Chandler http://imaginatorium.org
From: Virgil on 27 Jul 2005 23:51
In article <MPG.1d51aa051fa66820989fc2(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > For example, presumably there is some maximum length > > to the finite binary strings, which we will call L. > > How many binary strings are there then? 1 + 2 + 4 + ... + 2^L, > > which we all know is 2^(L+1)-1, which is finite, and is clearly > > larger than L (assuming L > 0). Now why someone would believe that > > there can exists 2^(L+1)-1 binary strings, but there cannot exist > > binary strings with length 2^(L+1)-1 is quite beyond me. They > > are both finite numbers. Why is the limit on finite string lengths > > smaller than the limit on finite sets of finite strings? > It's not Then TO has a contradiction on his hands, because as soon as the the length of a string can equal number of strings, we get a spiralling increase in both that has no finite limit. |