Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 25 Jul 2005 14:41 In article <MPG.1d4eb2755e6dd715989f65(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow <aeo6(a)cornell.edu> said: > > 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. > > Yes, I understand the difference between those two statements, and in > this case the two are equivalent. TO is off on his insane hobbyhorse of infinite in the sense of"endless" meaning that one must have an end to the endless. That is like saying that because time is endless, at least as far as we can tell, that time will eventually reach infinity. > If the length of strings is L and > the symbol set has a finite size of S, then you have S^L strings, > which is infinite IF AND ONLY IF L is infinite. Same false assumption that an infinite sequence must contain its limit. If a non-empty set of digit strings is finite, it will contain a longest string. Just line them up by length and take the last one. TO's claim requires that at some point adding one more character to a finite string makes it actually infinite. > Infinite Set <-> Infinite Element. TO <-> WRONG! > > If you wrote these out as logical statements, you would see that > > you are mixing up the order of quantifiers: > > > > (1) forall b, exists s in S, > > (if b is a finite bound, then length(s) > b) > > > > (2) exists s in S, forall b > > (if b is a finite bound, then length(s) > b) > > > > Statement (1) says that the *set* S has no finite bound. Statement > > (2) says that S contains an *element* that has no finite bound. > > Those are two different statements. > > I don't need a lesson in logic, thanks. You need something to break you out of your delusions, TO. > I see now why WM is > constantly accused of quantifier dyslexia. Because he cannot see the difference between "for all x there is a y such that ..." and "there is a y such that for all x ..." > It's because folks here > refuse to see what is abundantly clear and obvious, no matter how > carefully it is explained, or how many times. It's Cantorian > dyslexia. This point could not be more clear, and yet you seem almost > to insist that S^L can be infinite with finite S and L, No one is saying anything like that. What we ARE saying is that neither S nor L need be bounded by any finite value, so that S^L need not be bounded by any finite value either. Perhaps if TO ever bothered to read what we actually say, he would not misrepresent us so. > and that you > can increment a value an infinite number of times and still have a > finite value. Again a misrepresentation! What we actaully say is that there is no finite number of iterations at which we must stop, and so there is no finite limit on the size of naturals. Given any finite limit, there are naturals larger than that finite limit. > Can this really be the position of the professional > mathematical community? My statement is, TO's misrepresentation is not. Do you really not understand what "if and > only if" means? A good deal better than TO does, at least on the evidence he presents.
From: georgie on 25 Jul 2005 14:41 >Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic >theory. How can axioms be rigorous? Aren't they supposed to be self-evident? Doesn't that sort of imply that you "believe" them to be true? Isn't that faith?
From: imaginatorium on 25 Jul 2005 14:53 Tony Orlow (aeo6) wrote: > Daryl McCullough said: <snip> > > You are confusing two different things: > > > > (1) For every string, there is a finite number n such that > > the length of the string is less than n. > > > > (2) There is a finite number n such that for every string > > the length of the string is less than n. > > > > (1) is true, but (2) is false. > They are both false if you allow infinite strings. What makes you think i am > confusing those two statements? I am not. Well, yes, if you consider a set of strings which may be infinite strings (having only one end), then we all agree that both (1) and (2) are false. But the reason I'm sure I'm not alone in thinking you are confusing the two statements is that in the case in which one is true and the other is not, you appear to be unable to distinguish them, and use the truth of one permuted into the truth of the other. When we consider the set of finite strings, by definition of 'finite' ("the ditty stops") (1) is true, but if there is no limit on the length of the finite strings, (2) is false. Oh dear, I'm repeating someone else. Well, it's hopeless, isn't it? I suppose I felt prompted to respond because you claim you are not confusing (1) and (2). If you are not, you should be able to give us a different example where one is true and the other is false, no? (Otherwise you are not confusing them, because they are equivalent statements.) Brian Chandler http://imaginatorium.org
From: MoeBlee on 25 Jul 2005 14:54 georgie wrote: > >Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic > >theory. > > How can axioms be rigorous? Aren't they supposed to be self-evident? > Doesn't that sort of imply that you "believe" them to be true? Isn't > that faith? Axioms are rigorous in that there is an effective method by which to determine whether a formula is an axiom. Non-logical axioms are, by definition, true in some models and not true in others. Just to study a theory, one does not have to commit to a belief that a particular model is one of the real world, whatever one takes 'the real world' to mean.
From: Virgil on 25 Jul 2005 14:57
In article <MPG.1d4eb359702290ab989f66(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > >imaginatorium(a)despammed.com said: > > >> Tony has no clue what mathematics is, nor how it is done, so he doesn't > > >> normally bother with definitions. The closest we got from him for a > > >> definition of "finite" was that a finite number is less than an > > >> infinite one. And you can guess the "definition" of infinite. > > >Well, that's about as close to a lie as one can get, eh? > > > > >I asked for a definition of infinite, and no one could give me a > > >definition of that word. The best I could get was that an infinite > > >set can have a bijection with a proper subset, which is hardly a > > >definition of the word "infinite". > > > > On the contrary, that's a perfectly good definition of the concept > > "infinite set". > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > But not the word "infinite" on its own. Do you think the dictionary has the > word "set" in the definition of "infinite"? Are sets the only way to think > about infinity? Not hardly. We have also given a meaning to "finite natural" as an n in N for which the set {m in N: m <= n} is a finite set. This, of course, means that there are no infinite naturals, which TO does not like, but it is our definition, not his. For functions f:N -> R, we say that they converge to the finite limit L in R if and only if for every positive real epsilon (however small) the set {n in N: | f(n) - L | > epsilon } is finite. For functions f:N -> R, we say that they diverge to positive infinity if and only if for every positive real epsilon (however large) the set {n in N: f(n) < epsilon } is finite. For functions f:N -> R, we say that they diverge to negative infinity if and only if for every positive real epsilon (however large) the set {n in N: f(n) > -epsilon } is finite. Dictionaries, unless they are mathematical dictionaries, are not commonly precise about the mathemaical usage of words. The meaning of "infinite", in mathematics, is context sensitive. There is no meaning that is universal. Give us a precise context, and we can give a precise definition. |