Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 26 Jul 2005 10:06 Barb Knox said: > 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. > > Barb, you're not saying anything new. I have heard it all before. I am not drawing my conclusions in any such confused way, and the fact that you see it as such only reflects on your inability to read. I don't know about Herc, but I am neither conflating properties of elements to sets, or vice versa, without reason, or suffering from quanitier dyslexia, or whatever excuses you want to make for either not understanding or refusing to consider my logic. If I thought the way you are suggesting, then wouldn't I also be claiming that the reals in [0,1) must have infinite values too? They MUST have infinitely long digital representations in the infinite set, but the values are finite. However, when the required infinite digital representations are to the LEFT of the digital point, THEN you have infinite values. Get it? No? Kinda sad. When you generate the set of naturals by incrementing each to produce the next, and the set is infinite and the product of an infinite number of such increments, each producing a larger natural number, haven't you added an infinite number of 1's, and doesn't that produce an infinite sum, in the form of an infinite whole number? Can you take an infinite number of finite steps (without changing direction) and end up a finite distance from your origin? If you start out with a set {1}, is the set size, S, the same as the range of values, R, plus 1? Yep. If S=R+1, and we add element S+1 to the set, is S incremented at the same time R is incremented, and if we repeat this an infinite number of times to produce the naturals, can the set size EVER exceed the range of values by more than 1? No, you can't produce a set of naturals whose value range is less than the set size minus 1, so you can't have an infinite set with a finite range of values. Hope that makes you a little less sad, and a little less condescending. -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 10:08 Barb Knox said: > 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. > > No, that is not my argument at all. I just wrote a long response to this, and the news reader glitched, and I don't have time to write it again, but that is not at all my logic. Pay attention. -- Smiles, Tony
From: Han de Bruijn on 26 Jul 2005 10:21 David Kastrup wrote: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > >>Daryl McCullough wrote: >> >>>Let A_n be the set of all numbers between 0 and 2*n. Clearly, >>> limit n --> infinity A_n/n = 2 >>>but that doesn't mean that A_n approaches a set that is twice >>>as large as the naturals. >> >>Finally! That's *exactly* what boggles my mind! >> >>>Your notion of "set size" is inconsistent. >> >>Maybe. But it means that there is NO smooth transition from the >>finite to Cantor's "infinite". > > Of course not. > >>And this is precisely what anti-Cantorians find unacceptable. IMHO >>an "infinite" set cannot consistently have a "size". > > This is a _perfectly_ valid point of view (as opposed to the views of > those you sympathize with which are wildly inconsistent). Uh, uh. How do you know with which I symphathize? Do you keep a record of my responses to everybody? [ .. rest deleted .. ] OK. It seems that we finally have arrived somewhere. I have one final question, though. Is it "legal" (according to mainstream mathematics) to call a set "countable" if it can be brought in 1:1 correspondence with the naturals? And uncountable otherwise? Perhaps it's trivial for anyone, but I'm a bit at lost (due to those heated "Cantor's diagonal argument" threads I think :-) Han de Bruijn
From: Tony Orlow on 26 Jul 2005 10:29 Virgil said: > In article <MPG.1d489d7fea8af732989f60(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Virgil said: > > > In article <MPG.1d48308522352190989f3d(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > Barb Knox said: > > > > > In article > > > > > <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, > > > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > [snip] > > > > > > > > > > >Infinite whole numbers are required for an infinite set of > > > > > >whole numbers. > > > > > > > > > > Good grief -- shake the anti-Cantorian tree a little and out > > > > > drops a Phillite. Here's a clue: ALL whole numbers are finite. > > > > > Here's a (2nd-order) proof outline, using mathematical > > > > > induction (which I assume/hope you accept): > > > > > 0 is finite. If k is finite then k+1 is finite. Therefore > > > > > all natural numbers are finite. > > > > > > > > > > > > > > That's the standard inductive proof that is always used, and in > > > > fact, the ONLY proof I have ever seen of this "fact". Is there > > > > any other? I have three proofs that contradict this one. Do you > > > > have any others that support it? > > > > > > Unless TO has a definition of finiteness of naturals that makes the > > > above proof invalid, one valid proof is enough. > > > > I have explained the flaw in this proof, and it is met with confusion > > because none of you seems to appreciate the recursive nature of > > inductive proof. > > The inductive axiom shortcuts that recursion, which is the point of the > inductive axiom. It says that if the recursive step can be proved in > general, then it never need be applied recursively. > > If TO wishes to reject the inductive axiom, only then can he argue > recursion. What a load of bilge water! Accepting the axiom as a general rule does not mean one has to immediately forget about the lgical basis for the axiom. The underlying reason, outside of the axiomatic system, that this axiom holds true, is the transitive nature of logical implication, such that (a->b ^ b->c) -> (a->c). The construction of inductive proof produces an infinitely long chain of implications, throughout which the transitive property holds, so that the implication then applies to all members of the set. If you want to pretend that it's true because Peano said so, then you are just not thinking. > > > I am wasting my time with you, unless I write a > > complete elementary textbook. > > Please do, we can use the laughs. You do enough laughing. > > > > > > We have yet to see any of TO's alleged counter-proofs that are not > > > fatally flawed. > > > You have yet to point out any fatal flaw. > > That TO does not choose to acknowledge those flaws does not mean that > they are not there. That Virgil wants to claim they are there does not mean that this isn't just more dishonest garbage. If Virgil actually had any objection to the symbolic system one, then he would put it forth, and not shoot himself in the foot, as he did, below. > > > The best you have done is repeat your mantra of "no largest finite" > > on the inductive one, which is irrelevant. > > > Except to the issue at hand. If there is no largest finite natural then > the successor function on the naturals proves that the set of finite > naturals is infinite in the sense of the Cantor definition of infinite. Wouldn't it be nice if the "Cantor definition" agreed with ANYTHING else? > > And then there is no need for any of TO's alleged "infinite naturals". I have shown that there is, despite your repeated assertions to the contrary. > > > You have been mute on the information theory one, > > TO's "information theory" claim requires that at some point one can no > longer add another character to a character string and still have a > "finite" string. You obviously have not been paying attention. I never said anything even remotely like that. Learn to read. > > > and tried to claim there is no infinite sum of 1's in the infinite > > series one. > > Which addition of one to a prior natural carries over from finite to > infinite? "no largest finite. no largest finite...." Argument by irrelevance, anyone? > > > There is no fatal flaw that anyone has pointed out in my > > valid proofs. > > Willful blindness is not an adequate argument. Neither are empty statements and claims to victory. > > > Try addressing the situation, without making dishonest > > statement repeatedly conscerning my position or your achievements in > > refuting them. You're really a dishonest fellow, I must say. > > That is no more true than what TO mislabels proofs. Yes, very good. Respond with more ad hominems. A clear sign of a weak argument, or really, none at all. > > > > > > > > Inductive proof proves properties true for the entire set of > > > > naturals, right? > > > > > > > Wrong! It proves things only for the MEMBERS of that set, not the > > > set itself! > > And if a set is defined by each member with properties relating to > > that member, then those are all properties of that member. You have > > claimed repeatedly that I am making some sort of leap, and I have > > corrected you on that, and you failed to reply to those corrections, > > only to repeat your lies at a later time. Shut up and listen for a > > change. Maybe you'll learn something new for a change. > > > > > > Definitions (Cantor): (1) a set is finite if and only if there do > > > not exist any > > > injective mappings from the set to any proper subset > > > (2) a set is infinite if and only if there exists any > > > injection from the set to any proper subset. > > > Clearly then, a set is finite if and only if it is not infinite. > > > Definitions (Auxiliary): (3) a natural number, n, is finite if and > > > only if the set > > > of naturals up to it, {m in N: m <= n}, is finite > > > (4) a natural number, n, is infinite if and only if the set > > > of naturals up to it, {m in N: m <= n}, is infinite > > > > > > If these definitions are valid, then it is easy to prove buy > > > induction that there are no such things as infinite naturals: > > > > > > (a) The first natural is finite, since there is clearly no > > > injection from a one member set the empty set. > > > > > > (b) If any n in N is finite then n+1 is also finite. > > > This is also while quite clear, though a comprehensive proof > > > would involvev a lot of details. > > > > > > By the inductinve axiom, goven (a) and (b), EVERY MEMBER of N is > > > finite, but that does not say that N is finite. > > > > > N is finite if every member of N is finite. Show me how you get > > infinite S^L with finite S and L. > > 1^L + 2^L + 3^l + ... diverges, > S^1 + S^2 + S^3 + ... diverges for all S > 1. > > Unless TO can show that each of these has a finite limit, he is refuted. > I am tired of your foolishness, Virgil. You are creating infinities by combining, first, all the strings of length L from a set of 1 symbol, plus those from a set of 2 symbols, etc, up to an infinite set of symbols, and in the second, combining the set of strings from a set of symbols S of length 1, plus those of length 2, etc, up to infinite lengths. You proved my point in your refutation. The only way to get infinite S^L is infinite S or L, and S is finite in a digital number system, so L is infinite. You need infinite strings. Your refutation is refuted. You know how to deal with quicksand? -- Smiles, Tony
From: Robert Low on 26 Jul 2005 10:30
Tony Orlow (aeo6) wrote: > Barb, you're not saying anything new. I have heard it all before. I am not > drawing my conclusions in any such confused way, Differently confused, then. It's hard to tell. > If I > thought the way you are suggesting, then wouldn't I also be claiming that the > reals in [0,1) must have infinite values too? So, consider the rationals in [0,1). Each of them is (by definition) of the form p/q, where p is a natural number, q is a natural number other than 0, and p and q have no common factors. Are there rationals in [0,1) where p and/or q have to be infinite? |