Cantor Confusion
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From: David Marcus on 4 Nov 2006 11:54 mueckenh(a)rz.fh-augsburg.de wrote: > 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 da=DF omega die erste ganze Zahl sein soll, welche > > > auf alle Zahlen n folgt, d. h. gr=F6=DFer 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. That's not a definition. It is just a remark. All it says in modern terminology is that omega is the smallest ordinal larger than every natural number. No one disagrees with this. > > 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. You were asked for a definition of "1/omega" and you state something that is not a definition and which is meaningless without a definition of "1/omega". > Now replace 2 by n and let n-->oo. Still meaningless without a definition of "1/omega". -- David Marcus
From: David Marcus on 4 Nov 2006 11:59 mueckenh(a)rz.fh-augsburg.de wrote: > 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. Moe was discussing axiomatic set theory, specifically ZF without the axiom of infinity. You clearly are discussing philosophy, e.g., "greatest errors of the human mind". If you wish to discuss philosophy, feel free, but please do not inject philosophy into a discussion of mathematics without saying you are doing so. > > 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. That what was not? Modern mathematics does not use the term "finished infinity". If you wish to use it, please define it. > > > 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. Usually, the word "read" means read and understand. You haven't demonstrated any understanding of the book. -- David Marcus
From: David Marcus on 4 Nov 2006 12:03 mueckenh(a)rz.fh-augsburg.de wrote: > 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? Because you don't state clearly how you wish to divide the edges and how the division is used in the "relation" between edges and paths that you are defining. > > > 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. Regardless of whose terminology it is, if you use it, you should be willing to define it when asked. So, please define what you mean by "whole number". > Please don't conclude from your own ignorance on mine. Wouldn't dream of it. -- David Marcus
From: David Marcus on 4 Nov 2006 12:14 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > In article <1162472271.664965.315880(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > Does WM choose to claim that a set of naturals which has no maximum must > > > still have a finite upper bound? > > 1 > 11 > 111 > ... > > The length of each column is omega. I assume that we are using ZFC as our logical system. Assuming by the "length of each column" that you mean the number of 1's in each column, then the number of 1's in each column has cardinality aleph_0. So, what you wrote is essentially correct. > The length of each line is less than omega. Each line has a finite number of 1's. Finite cardinality is less than aleph_0. So, what you wrote is essentially correct. > The length o the diagonal is less than omega. The number of 1's in the diagonal is aleph_0. So, what you wrote is false. In fact, I have no clue what you could possibly be thinking that would lead you to make such an obviously incorrect statement. The length of the diagonal is clearly the same as the length of the first column, and you just wrote above that the length of each column is omega. > Set theory is based upon intermingling maximum and supremum. If what you wrote above is supposed to demonstrate this, then you have failed for two reasons: You made a false statement, and you didn't use the words "maximum" or "supremum" in what you wrote. -- David Marcus
From: Lester Zick on 4 Nov 2006 12:29
On Fri, 3 Nov 2006 17:19:15 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Thu, 2 Nov 2006 18:57:30 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >Virgil wrote: > >> >> WM merely repeated his automatic error several more times here. >> >> >> >> WM claims that a list in which the nth listed element is a string of >> >> length at least n characters cannot produce a diagonal of length >> >> greater that any finite number of characters. >> >> >> >> His claim is trivially and obviously false, but he keeps repeating it ad >> >> nauseam, as if by sufficient repetition of that lie , he can make the >> >> truth go away. >> > >> >Not only does he keep repeating it, but he never even tries to justify >> >it in any way. He is like a broken record. We ask him for his reason and >> >he just repeats the same unjustified, erroneous claim. Kind of boring. >> >> No more boring than endless repetition of "true" "infinity" "set >> theory" and "bijection". Kinda grates on the nerves. Repetition and >> truth just don't converge. > >Are you referring to your own posts? I'm referring to the posts of those who biject set theory with truth. ~v~~ |