Cantor Confusion
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From: mueckenh on 20 Nov 2006 10:18 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> >> >> The cardinality of omega is |omega| not omega. [1] > >> >> > > >> >> > Kunen's "Set Theory" defines |A| to be the least ordinal that > >> >> > can be bijected with A. So, with this definition, |omega| = > >> >> > omega. > >> >> > >> >> You are absolutely right. > >> > > >> > Well learned (after all)! > >> > > >> > Now try to understand the next step: > >> > > >> > If omega exists, then |omega| =/= omega & |omega| = omega. > >> > >> Sorry how did you arrive at > >> > >> |omega| =/= omega (*) > > > > Look a the first line [1], written by yourself. > > <456068f6$0$97216$892e7fe2(a)authen.yellow.readfreenews.net> I cannot understand your explanation given there. If you say "The cardinality of omega is |omega| not omega", so you must have had in mind |omega| =/= omega, which is wrong, (according to the late Cantor - you see it is useful to study him). Would it be meaningful to write "Y is X but not X"? Regards, WM
From: mueckenh on 20 Nov 2006 10:34 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > > > The axiom of infinity says that the set N exists. Your iia says > > > > > > > > > > "we cannot recognize or treat all of its elements" > > > > > > > > This is no contradiction to the axiom. Compare the proof that the real > > > > numbers can be well-ordered. We cannot construct or define or > > > > recognize a well ordering. > > > > > > The fact that we cannot construct a real ordering does not > > > stop us from proving such an ordering exists > > > > LOL. I know. > > > > > and using > > > the existence of such an ordering. > > > > The existence of a well-order does not guarantee the constructibility > > or definability of a real ordering. > > The existence of a set does not guarantee the existence or definability > > of a bijection or an identity mapping. > > No. > > To say that a set N exists is to say that all elements n of N exist. > Thus the mapping n ->n for all elements n of N exists. That is exaggerated. There are models of ZFC (including countably many elements a part of which could be interpreted as the set of natural numbers) where no mapping on N exists. Why should it fail if you were right? > > > > > > > > > > >And even if we could, the > > > > axiom of infinity does not prove that infinite ordinal and cardinal > > > > numbers are numbers, i.e., that they stand in trichotomy with each > > > > other. > > > > > > Check the definitions of ordinal and cardinal below. Look for > > > the term trichotomy. Fail to find the term trichotomy. Draw > > > the obvious conclusion. > > > > My conclusion is: You don't know this term. > > No, this is not the obvious conclusion (it is also wrong). > The obvious conclusion is that the > fact that the term tricotomy is not used when defining > either the ordinals or the cardinals, means that we > do not need to show trichotomy to show that > something is an ordinal or a cardinal. It may be called by another name but trichotomy is implied. In fact all ordinals are asserted to stand in trichotomy with each other. > > > Nevertheless, you will > > agree if you learn what trichotomy means: a < b or a = b or a > b for > > any two numbers a and b. > > > > Trichotomy is a property of ordered sets, not of sets. Only ordered sets have ordinal numbers / are ordinal numbers. > We need to put > an ordering on the ordinals or the cardinals before we can say > whether trichotomy holds. Ordinals are ordered sets by definition. We need not put anything. > If we put the natural ordering on the > ordinals, then trichotomy holds for all ordinals (finite and infinite). Yes, why then do you dislike the name? > > > Piffle. If the natural numbers exist, then the identity map is > > > a bijection. > > > > If it exists, of course. > > > > Piffle. If the set of natural numbers exists then the identiy map > on the natural numbers exists. > You are wrong. But this carelessness is typical for set theory. Regards, WM
From: mueckenh on 20 Nov 2006 10:46 Sara M schrieb: > > But the set of all lists is countable (as is any quantized or > > discontinuous set), so is the set of all list entries. > > That's absurd. First of all, I'm not really clear what "discontinous" > means when applied to a set, Why then do you call it absurd? Here are some explanations: A list is a (injective) sequence. Discontinuous means: The difference between two elements (i.e. natural numbers enumerating a list) has a finite minimum value. Every list occupies a non vanishing part of the space of he universe. The volume of the (accessible) universe is finite. Regards, WM
From: Franziska Neugebauer on 20 Nov 2006 10:59 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> >> >> The cardinality of omega is |omega| not omega. [1] >> >> >> > >> >> >> > Kunen's "Set Theory" defines |A| to be the least ordinal that >> >> >> > can be bijected with A. So, with this definition, |omega| = >> >> >> > omega. >> >> >> >> >> >> You are absolutely right. >> >> > >> >> > Well learned (after all)! >> >> > >> >> > Now try to understand the next step: >> >> > >> >> > If omega exists, then |omega| =/= omega & |omega| = omega. >> >> >> >> Sorry how did you arrive at >> >> >> >> |omega| =/= omega (*) >> > >> > Look a the first line [1], written by yourself. >> >> <456068f6$0$97216$892e7fe2(a)authen.yellow.readfreenews.net> > > I cannot understand your explanation given there. If you say "The > cardinality of omega is |omega| not omega", so you must have had in > mind |omega| =/= omega, No Way! If you want to misapprehend me do so, but don't confuse your misapprehensions with theorems of set theory. > which is wrong, (according to the late Cantor - you see it is useful > to study him). Cantor is not generally normative on these issues as one of the "proponents" has already pointed out. Under a common definition of cardinal number omega is the cardinal number of omega. There is no doubt about it. Since you persistently refuse to give defintions wouldn't you call me presumptous if ever I uttered what "you must have had in mind"? > Would it be meaningful to write "Y is X but not X"? This is not a "faithful" representation of my sentence you have objected to because there is no character string which can be substituted for X to get my original sentence back. You seem to like removing vertical bars (and braces elsewhere) at will. F. N. -- xyz
From: mueckenh on 20 Nov 2006 11:06
William Hughes schrieb: > > You are trying to argue > > The diagonal cannot be longer than every > initial segment with a largest element because > there is no set of lines without a largest line. No. The diagonal cannot be longer than every line because it consists of line-ends only. > > There is no set of lines without a largest line > because the diagonal cannot be longer than > every initial segment with a largest element. > > This is circular. > No. Again the arguing is: The diagonal cannot be longer than every line because it consists of line-ends only. If the diagonal has omega elements, then a line must have omega elements. if no line has omega elements, then the diagonal cannot have omega elements. This is not circular but a bijection, the same bijection as between first column and diagonal. The bijection between lines and initial segments of the first column is brought about by the diagonal. The reason for this bijection is the fact that natural numbers count themselves. 1 2 3 .... <--> 1,2,3...n This bijection fails in case of the full column with omega elements but no line with omega elements. That is so obvious that I cannot understand how one can miss it. Regards, WM |