Obections to Cantor's Theory (Wikipedia article)
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From: Robert Low on 26 Jul 2005 17:37 Tony Orlow (aeo6) wrote: > Robert Low said: >>Tony Orlow (aeo6) wrote: >>>There ARE no infinite numbers, only infinite sets, in your >>>lexicon. >>Not only do we know about infinite numbers, but there are >>at least two different kinds of them: cardinal and ordinal ones. > So, you think there can be infinite whole numbers, but that they are not in the > infinite set of whole numbers? I didn't say that there were infinite *natural* numbers; I said that there were at least two different types of infinite number, namely cardinal and ordinal ones. The finite ordinals are the natural numbers, and the set of all finite ordinals is an infinite set each of whose elements is itself finite. > Why am I being challenged on the very notion of > such numbers, if their existence is already established? You are being challenged on the notion of 'infinite natural numbers' as necessarily existing in the set of all standard natural numbers, because that's poppycock. There *are* models of the (first order) Peano Axioms which contain objects you could call infinite integers. These models even have some of the properties you seem to want to ascribe to the natural numbers. These models are called non-standard precisely to distinguish them from the usual, or standard model which does not contain such integers. I suspect you'd find this stuff interesting if you bothered to learn about it, and you might even come to the conclusion that one of these non-standard models is the 'right' model of the natural numbers. If so, you should read up on non-standard analysis, and internal set theory in particular. But the debate between the mainstream approach and that one is not that only one of the approaches is consistent, rather that one approach is more useful than the other.
From: Virgil on 26 Jul 2005 17:39 In article <MPG.1d506423e45b1407989f9a(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Robert Low said: > > Tony Orlow (aeo6) wrote: > > > The problem comes whenever someone wants to use the word "infinite" in > > > any > > > other context. > > > > The problem comes when people use a word in one context as if > > its meaning were specified by a different context. That's what > > an amphiboly is. > > > > > There ARE no infinite numbers, only infinite sets, in your > > > lexicon. > > > > Not only do we know about infinite numbers, but there are > > at least two different kinds of them: cardinal and ordinal ones. > > (Still other notions of infinite number occur in other contexts.) > > Again, clear definition of what is meant in a particular > > context is required. > > > > > Talk about needing to learn about language. What is so hard about > > > infinite numbers, or trying to define the word itself, independent of > > > bijection > > > genuflection? > > > > What is so hard about the idea of context sensitivity, and accepting > > that the word 'infinite' can mean different things in different > > contexts? Does it bother you than 'bread' can refer to products > > made from different grains, or can even refer to 'money', depending > > on the context? > > > So, you think there can be infinite whole numbers, but that they are not in > the > infinite set of whole numbers? Why am I being challenged on the very notion > of > such numbers, if their existence is already established? No one has "established" that any infinite naturals exist, and as TO's "whole numbers" are as yet undefined, the isssue of whether they can be infinite is moot.
From: Daryl McCullough on 26 Jul 2005 17:29 Tony Orlow says... > >Chris Menzel said: >> Oh please. There isn't just a difference in points of view here. TO is >> making demonstrable, elementary mistakes of both logic and mathematics. >Untrue. What Chris says certainly is true. Your claim that every infinite set of string must contain strings of infinite length is just false. It's not a matter of opinion. It's not a matter of definition. It's just wrong. A Turing machine programmed to print out every finite bit string will never halt (because it is an infinite set), and it will never print out an infinite bit string (because Turing machines can't print out any infinite string). >Bijection between infinite sets as proving equal size. I disagree with that >assumption. It's not an assumption. It is a *definition*. You can say that you are uninterested in the Cantorian definition of set size, but to say that you disagree with it is just nonsensical. -- Daryl McCullough Ithaca, NY
From: Virgil on 26 Jul 2005 17:48 In article <MPG.1d506874c11633d0989f9b(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow <aeo6(a)cornell.edu> said: > > > > >Daryl McCullough 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. > No, I misread those statements. The first, if you are referring to my > arguments, should be that there is an infinite set of whole numbers, and the > second is that there is an infinite whole number in the set. How do TO's "whole numbers" differ rom "natural numbers"? Until we have some idea of what TO means by "whole numbers", this whole discussion is pointless. And if TO means the same thing as natural numbers he is wrong. Those two > statements imply each other because of the constant finite difference between > whole numbers. This has nothing to do with quantifier dyslexia or other > logical > errors on my part. It has precisely to do with quantifier dyslexia and other problems on TO's part. > > > > >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. Infinite Set <-> Infinite Element. > > > > You've just contradicted yourself. I thought you agreed that > > there is a difference between the two sentences > > > > (1) There is no finite bound on the lengths of strings in S. > > (2) There is a string in S that is infinite in length. > > > > In case (1), the set is infinite, but there is no infinite element. > (sigh) > There is a difference between those two statements. They are not equivalent. > The two I just made, which you think contradict something else I said, ARE > equivalent for whole numbers. I never said (no finite bound)<->(infinite). I > said (infinite set size)<->(infinite whole number element) for natural > numbers. If "whole number" means "natural number", TO is wrong. In any case, there in no unbounded member of N, whatever those members are called. Every member of N is smaller that some finite member of N. > > > > You seem to be thinking that there is some L characterizing the > > lengths of strings. There isn't. There are strings of length 1, > > there are strings of length 2. There are strings of length 3. > > Each string has a length, but there is no length associated with > > the set of finite strings as a *whole*. > No, but if L can ONLY be finite, then S^L can ONLY be finite. If you do not > allow L to be infinite, then you do not have infinite S^L. > > > > >> 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. That you do not know that you need such lessons is part of your problem.
From: Virgil on 26 Jul 2005 17:50
In article <MPG.1d5068c711a903e1989f9c(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow <aeo6(a)cornell.edu> said: > > > > >I don't need a lesson in logic, thanks. > > > > Yes, Tony. That is exactly what you need. You seem to be trying > > to learn some elementary facts about logic and mathematics by > > arguing on the internet. That isn't a very efficient way for a > > beginner to learn. If you're at Cornell, then go take a course > > in logic. It's an excellent institution. > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > I majored in Computer Science, and took plenty of logic and discrete > math, and did quite well, thank you. Your inability to follow my > argument is not an indication of my logical ineptitude. We can many of us follow TO's arguments well enough to find their many faults, however often TO may deny them. |