Cantor Confusion
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From: Virgil on 3 Nov 2006 21:58 In article <1162564439.869681.94620(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > No. I believe that the length of the diagonal is the supremum of > > the length of the lines. This supremum does not have to be the > > length of any line. Call it omega. Then omega is the *supremum* > > of the set of lines (equally the supremum of the set of columns). > > The set of lines has infinitely many elements, no line has an infinite > number of columns. That is a difference. But a trivial one, as no line is infinitely far down the list of lines any more that any digit is infinitely far over from the beginning of its line. > > If you consider omega to be not the maximum but only the supremum of > the set of lines, then we agree that actual infinty does not exist. Wrong. as usual. If the set of lines, with no maximum, exists that it is an actually infinite set. So that WM must be agreeing that actual infinity DOES exist. > But we both then disagree with ZF which states *there exits* the set > of lines. I do not agree that that it went anywhere.
From: William Hughes on 3 Nov 2006 23:27 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > No. I believe that the length of the diagonal is the supremum of > > the length of the lines. This supremum does not have to be the > > length of any line. Call it omega. Then omega is the *supremum* > > of the set of lines (equally the supremum of the set of columns). > > The set of lines has infinitely many elements, no line has an infinite > number of columns. That is a difference. So? Two different sets can have the same supremum. > > If you consider omega to be not the maximum but only the supremum of > the set of lines, then we agree that actual infinty does not exist. No, the supremum exists and the supremum is actual infinity. > But > we both then disagree with ZF which states *there exits* the set of > lines. They all are there. And there does not exist a set of infinitely > many columns (= an infinite natural number) No, there does exist a set of infinitely many columns. However, this set is not a natural number. It is the supremum of a set of natural numbers. A supremum of a set A does not have to be a member of A. So while this set is an infinite number it is not a natural number. - William Hughes
From: mueckenh on 4 Nov 2006 07:37 MoeBlee schrieb: > As > to set theory, for the tenth time: Without the axiom of infinity it is > UNDETERMINED whether every set is finite. For the eleventh time: Infinity is NOWHERE. To assume the existence of actual infinity is one of the greatest errors of human mind. Therefore, without explicitly assuming this notion, it cannot be anywhere. > > > Whatever your point, you won't be able to show that merely dropping the > axiom of infinity from the Z axioms entails that there are only finite > sets. Whatever your point, you won't be able to show that merely dropping the axiom of rabbithood from the Z axioms entails that there are only non-rabbit sets. To put it in other words: It simply an imbecile nonsense to talk about finished infinity without explicitly stating that it was not. > > Modern set theory simply cannot describe developing sets as > > it apparently cannot describe sets with limited contents of > > information. > > Whatever your definition of "developing sets" and 'limited contents of > information", the fact remains that dropping the axiom of infinity does > NOT entail that there are no infinite sets. > > > These things are unknown to the slaves of formalism. Read a good book > > like Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: > > "Foundations of Set Theory", North Holland, Amsterdam (1984). There you > > will find more about that topic. > > Oh please, I've read more in that book than you have. That is strange. I read all of it, but you read more. This simple sentence alone would prove that you must be a set-theorist. Regards, WM
From: mueckenh on 4 Nov 2006 07:39 MoeBlee schrieb: > > Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu > > denken, welcher die Zahlen n zustreben, wenn darunter nichts anderes > > verstanden wird, als daß omega die erste ganze Zahl sein soll, welche > > auf alle Zahlen n folgt, d. h. größer zu nennen ist als jede der > > Zahlen n. (p. 195) > > > > This definition has been conserved up to our days: Limesordinalzahl or > > limit ordinal number. > > I regret that I don't read German. A disadvantage for a set theorist. Here is my translation: It is allowed to understand the new number omega as limit to which the (natural) numbers n grow, if by that we understand nothing else than: omega shall be the first whole number which follows upon all numbers n, i.e., which is to be called larger than each of the numbers n. > But I'd like to know what you might > propose as a mathematical definition of 1/omega in Z set theory. > Notice, I'm not asking what Cantor wrote. I'm asking what is the > definition of 1/omega specifically in Z set theory. If omega is larger than 2, then 1/omega is smaller than 1/2. Now replace 2 by n and let n-->oo. Regards, WM
From: mueckenh on 4 Nov 2006 07:41
MoeBlee schrieb: > For ordinals, > > x<y <-> xey > > where 'e' is the epsilon membership symbol. That is identical with my definition. Take for instance Zermelo's definition of the naturals or Cantor's own (Collected works, p. 289-290), then you can see it. Your definition has been simply translated from Cantor's. Please don't conclude from your own ignorance on mine. > > > If it cannot be a fraction because ZF does > > not yet know how to divide elements, > > In ZF we define various operations of division. As far as I know, there > is not a dvision operation for sets in general. Why then do you not understand how an edge of the binary tree can be divided? > > > then it can only be a whole > > number, I would guess.# > > These problems you're having are of fitting set theory to your own > system of terminology. To work in set theory, we don't need to care > about your own system of terminology. "Whole number" is not my terminology. Please don't conclude from your own ignorance on mine. Thank you. Regards, WM |