Obections to Cantor's Theory (Wikipedia article)
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From: Barb Knox on 21 Jul 2005 00:30 In article <slrnddto80.1o98.cmenzel(a)philebus.tamu.edu>, Chris Menzel <cmenzel(a)remove-this.tamu.edu> wrote: >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. A third aspect is the implict view that if some property holds for EVERY ELEMENT of a set then it also holds for THE SET itself. ("Herc" is particularly prone to this one.) Such a view would account TO's belief that if every natural number were finite then the whole set would also be finite, and hence that there must be at least one infinite natural. -- --------------------------- | 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 | -----------------------------
From: Virgil on 21 Jul 2005 00:33 In article <MPG.1d487f6a11e3fb9c989f5a(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Chris Menzel said: > > > > > > 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. > > > > In mathematics "think you know" is not good enough. When asked for > > precise and comprehensive definitions in mathematics, it is the > > duty of the user of any terms to give such precise and > > comprehensive definitions. > > > > > They are all widely understood. > > > > Not by me or by Chis. I can think of at least two antithetical > > meanings for each of TO's vague terms, so To piles ambiguity on > > ambiguity. > > > > > 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, without defining anything themselves, > > > and assuming unfounded postulates such as all infinities are the > > > same, except when they're not. > > > > Every bit of "Cantorianism" has been well enough defined for the > > understanding of thousands upon thousands of people. That TO fails > > where so many have succeeded says more about TO than about the > > adequacy of "Cantorianism's" explanations. > > > Why don't you try asking for clarification of a term if you don't > understand it? I have and have been refused on grounds that the meanings were obvious. > You are playing lawyer's interrogatory with me. I am playing a mathematician's interrogatory, trying to find out what TO is talking about, but as far as I can tell TO doesn't know either. > I am > not being sucked into that any more. Ask specific questions, insead > of trying to pretend I am saying the "kerfloppen is infinitely > gnudlio". That is at least as sensible as most of what TO says.
From: Virgil on 21 Jul 2005 00:43 In article <MPG.1d4880662f52666989f5b(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > Cantorians seem to think infinity is simply infinity, > > > > > > > > Hard to fault them on that score. The fact that you have > > > > doubts about whether infinity is infinity (what does "simply" > > > > add here?) says quite a lot -- though *perhaps* if you were to > > > > try to distinguish different senses of "infinite" with any sort > > > > of mathematical precision there might be a point there. > > > > > That was a dishonest snip, really. Notice the comma? That means > > > the sentence was not finished. The rest of it said that you do > > > this, even in the midst of proving that it is not the case. > > > > There is a difference between even naturals and odd naturals, but > > there are also differences between one even number and another. > > > > So to say that there are differences between infinite sets is > > equivalent to saying that there are different even naturals. > > > > > > > The diagonal argument assumes that the list is essentially > > > square, > > > > Geometry does not enter into it. Such a "list" assumes that for > > every natural there is a real. Decimal reprsentation assumes that > > for every real and every natural, n, there is an nth decimal digit > > to the right of the the decimal point for that real. > > > > > and can be traversed diagonally, but then proves that this is not > > > the case, and yet, Cantorians still seem to insist that the > > > diagonalization proves the reals cannot be listed, when what it > > > proves is that there are more reals than naturals. > > > > The whole point is to prove that there are more reals than > > naturals. If it does that, nothing else is needed. > > > If that's all you claimed, then I wouldn't be arguing, but that > larger infinity is placed in some different class of "uncountables", > and all "countables" are declared erroneously equal. All infinite countables are, by definition, bijectable to each other and to N, as that is what infinite and countable is defined to mean. If TO can't learn the meanings of these words, he should stop trying to use them. > That's where I diverge from the standard. And begin messing up. TO does not undersatand that thousands of people much brighter than he have worked these things out much better than he could ever hope to. > besides, while there are certainly more > reals than wholes, as can be clearly seen on the number line without > all this bijectional blooplah, it is not necessarily the case that > there are more reals between any two whole numbers than there are > whole numbers in general. You could just as easily have the reals > using half as many digits, or any other smaller infinity. The > interpretation of the result is what is really at issue here. One issue here is that TO keeps ignoring standard mathematical definitions, however often presented, and then declaring that that the defined words and phrases must have other meanings than the ones mathematicians have agreed on.
From: Virgil on 21 Jul 2005 00:51 In article <MPG.1d4881fed7bf903e989f5d(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d4825defff809ee989f39(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > I proved it for all n in N, which I think you agree is an > > > infinite number of naturals. > > > > But that does not prove that what is true for members of N is true > > for N itself. Consider the set containing a single apple. By TO's > > argument, that set must itself be an apple, since that is true for > > all its members. > > > > To uses a similar argument in reverse, that an infinite set must > > contain infinite objects, so not that apple must be a set as well > > as the set being an apple, at least by TO_logic. > You are deliberately misrepresenting what I have said, Virgil, as you > have done countless times. Stop it. You are lying. No, I am just using analogy to show how foolish TO's arguments are. > > > > > I think you were perhaps one of those claiming that inductive > > > proof only works for finite iterations, but then it wouldn't work > > > for the infinite set of naturals, now, would it? I am not going > > > in these stupid circles with you. > > > > TO makes up his own stupid circles, and goes round and round them > > endlessly. > So says the pot. So says the circler. > > > > > > Again this goes back to the Cantorian mantra, "no largest finite" > > > > It is not strictly Cantorian. To the best of my knowledge, even > > those mathematicians who avoid infinities do not object to 'no > > largest finite natural', though all mathematicians object to > > infinite naturals, at least in standard models. > I never objected to that fact either. It's just not an excuse for > dismissing obvious proofs. When obvious proofs are presented, they will not be dismissed, but those things that TO presents have not yet qualified. > > > > If TO wants infinite naturals, he had better look up Abraham > > Robinson, et al, but if he thinks Cantorian math is confusing, he > > will not get far with non-standard analysis. > >
From: Virgil on 21 Jul 2005 01:00
In article <MPG.1d489a75c7016d8e989f5e(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > >So, inductive proof does not rely on proving for n+1 based on n? > > >The infinite number of successive naturals for which you prove > > >your property do not constitute an infinite number of implied > > >steps in inductive proof? > > > > As has been pointed out before, an inductive proof does not have an > > infinite number of steps. To prove "for all natural numbers x, > > Phi(x)", you only need to prove the following two statements: > > > > 1. Phi(0). 2. for all natural numbers x, Phi(x) implies > > Phi(x+1). > > > > You seem to be thinking that proving statement 2 somehow requires > > an infinite number of steps. If that's the case, then statement 2 > > doesn't *have* a proof (because proofs have to be finite). > > > > A proof of a universal statement is not the concatenation of > > infinitely many singular proofs. > > > > -- Daryl McCullough Ithaca, NY > > > > > It seems to me you are all being dense and objecting to something I > am not saying. I hope that's not the case. > > The proof has a finite form, much like a recursive algorithm. A > recursive algorithm will run forever if it doesn't have some stop > condition, like running out of nodes in a tree path, which is bad for > a computer program. But this recursive proof, like the recursive > definition of the naturals, DOES go on forever. Then TO misses the point entirely. The inductive axiom itself is what takes it out of the recursive class. And since it is an axiom, it needs no proof. TO is permitted to reject that axiom, or even the entire axiom system, but, having done so, he is not empowered to criticize the deductions made within a system which he rejects. Your #2 above is the > recursive part of the proof; it proves something true based on the > truth of its predecessor. When you actually "run" the proof, #2 acts > like a loop, iterating its way through all the members of the set, > with no stop condition. The axiom shortcuts that looping. |