Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 25 Jul 2005 18:39 In article <MPG.1d4f02d248d9b922989f78(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > >> "Bigger" in the sense of no surjection from the "smaller set to the > > >> "larger", is one thing, "bigger" in the sense of having the "smaller" > > >> set as a proper subset is different. While these two measures happen to > > >> coincide for finite sets, they do not coincide for infinite sets, as the > > >> definition of infinite for sets should hint to you. > > >> > > >gee, they coincide for finite sets AND infinite sets, under Bigulosity > > >but I don't suppose you consider that extra consistency any sort of > > >progress. > > > > No, that's no progress at all. You can prove that for finite sets > > if there is a bijection between set A and set B, then they have > > the same Bigulosity. But that fails for infinite sets. So Bigulosity > > is an inconsistent notion of "size". > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > Huh? Bigulosity doesn't rely on bijections. That why it DOES work. For finite sets bijection works much more easily than bigulosity, and I have seen no evidence that it works any less efficiently or accurately that bigulosity for infinite sets. Unsupported claims by the inventor of bigulosity do not convince.
From: Robert Low on 25 Jul 2005 18:41 Virgil wrote: [re TO] > If he wants to make up his own game, fine, but he will not find many > willing to play by his rules. He'll have to tell us what the rules are before we can decide if the game is interesting.
From: Virgil on 25 Jul 2005 18:48 In article <MPG.1d4f04317d22ce87989f79(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Chris Menzel said: > > On Wed, 20 Jul 2005 20:55:58 +0100, Robert Low <mtx014(a)coventry.ac.uk> > > said: > > > Daryl McCullough wrote: > > >> That's not true. If S is an infinite set of strings, then there is a > > >> difference between (1) There is no finite bound on the lengths of > > >> strings in S. (2) There is a string in S that is infinite. > > > > > > Except that TO claims that (1) implies (2), though I can't > > > even get far enough into his head to see why he thinks it, > > > never mind finding his 'argument' convincing. > > > > This seems to be a fairly common element in crankitude. I've seen > > several folks argue here and elsewhere that there can be infinitely many > > natural numbers only if there is an infinite natural number. (Indeed, I > > think TO believes this, as I believe I saw reference to an "infinite > > natural" in one of his posts.) The origin of this idea sometimes seems > > to reside in imagination -- the poor afflicted fellows picture the > > number sequence as something like an endless string of beads that > > eventually disappears to nothing. There is thus no perceptual > > difference between *really really long* proper initial segments of the > > string and the entire string itself. So the entire string is the same > > sort of thing as its really really long proper initial segments. Other > > times, there seems to be some sort of a priori cardinality principle at > > work: for every set of natural numbers there is a natural number that > > numbers them. No finite natural number numbers all the finite natural > > numbers, so (obviously) there is an infinite natural number. > > > > Whatever. Kinda sad. > > > > > What is sad is that you say you have heard the same arguments repeatedly, and > still don't seem to understand them. We understand them, we just don't believe them. If they were true, endless sequences that diverge upwards would actually have to contain an infinite value instead of merely being unbounded. Consider the endless seqeuence of reals, n -> n+1/n. According to TO, there must be a natural number, n, for which n+1/n is actually infinite, but according to standard limit theory all we need to know is that for every positive epsilon, howwever large, the set {n in N: n+1/n <= epsilon} is finite.
From: Lee Rudolph on 25 Jul 2005 18:49 Robert Low <mtx014(a)coventry.ac.uk> writes: >Virgil wrote: >[re TO] >> If he wants to make up his own game, fine, but he will not find many >> willing to play by his rules. > >He'll have to tell us what the rules are before we can >decide if the game is interesting. At this point (and I think long before this point), we have a lot of evidence that he is one or more of (a) irremediably stupid, (2) hopelessly mad, (iii) successfully trolling. In each case, it's unlikely that the game is going to be interesting; except, indeed, to those (who seem to be many) who enjoy spending their summers (if in the Northern Hemisphere) dealing with members of one or more of those classes. Lee Rudolph
From: Barb Knox on 25 Jul 2005 18:51
In article <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: [snip] >keep in mind that >inductive proof IS an infinite loop, so that incrementing in the loop creates >infinite values, and the quality of finiteness is not maintained over those >infinite iterations of the loop. Using your computational view, consider the following infinite loop (using some unbounded-precision arithmetic system similar to java.math.BigInteger): for (i = 0; ; i++) { println(i); } Now, although this is an INFINITE loop, every value printed will be FINITE. Right? So, now can you see that the INFINITE set of natural numbers has only FINITE elements? [snip] -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- |