Cantor Confusion
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From: mueckenh on 21 Nov 2006 16:00 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > There are two types of initial segments > > > > > > Initial segments with a largest element > > > Initial segments without a largest element. > > > > > > There is an initial segment of the diagonal that does > > > not have a largest element. > > > (Recall: the diagonal has a largest element if and only > > > if there is a last line. There is no last line.) > > > Every line has a largest element. Thus there is an initial > > > segment of the diagonal that is not a line. > > > > The diagonal has only finite indexes n. Therefore it consists of line > > ends only. Every line end is the end of a line, no? > > > A classic case of > > > Person 1: > > P_1 is true. <reasons why P_1 is true> > P_1 implies P_2 > Therefore P_2 is true > > Person 2: > > No > P_1 is true > However, P_1 does not imply P_2 <reasons why P_1 does > not imply P_2> > P_2 is false > > > Person 1: > > Clearly you do not understand. > Here is another reason why P_1 must be true. > > > Yes. P_1: The diagonal consists of the ends of lines. > > No. The fact that > P_1: the diagonal consists of the ends of lines > does not imply that > P_2: the diagonal is not longer than every line. > > > > > > > > Therefore you cannot say: > > > > > > The diagonal cannot be longer than every line > > > because it consists of line-ends only. > > > > Of course that must be true. The diagonal has only finite indexes n. > > Every line end is the end of a finite line. > > As it cannot be infinite, the complete diagonal does not exist. > > > > > > > If element d_nn of the diagonal is a member of line n, then all > > elements d_mm wit m =< n of the diagonal are members of the same line > > n. In other words: There is never more than one single line required to > > establish a bijection with all the elements of the diagonal d_mm with m > > =< n. > Yes. And the diagonal consists of only such elements. > > However, since there is always an element (n+1), no single > line can establish a bijection with *all* the elements of the diagonal > (both those <=n and those > n). Together with the proof above this shows that "not *all* the elements of the diagonal" exist. > > > > As the diagonal has only finite indexes n, there is never more > > than one line required to establish a bijection with all elements of > > the diagonal. > > No. No line with finite index n can establish a bijection with > all elements of the diagonal. Every line has a finite index. > Therefore no line can establish a bijection with all elements of the > diagonal If there is a bijection with all finite line ends, then there is a bijection with one line. Why do you prefer one of them? Regards, WM
From: mueckenh on 21 Nov 2006 16:05 William Hughes schrieb: > > What about all contructible numbers or all words? They cannot be mapped > > on N though they are a countable set. > > > Absolute piffle. The question is whether the set N can > be mapped to the set N, not whether some other > set can be mapped to the set N. It is the same impossibility, but not so obvious. > > [Although it is interesting to note that both the constructable numbers > and the set of all words can be mapped to the set N (please check > the definiton of countable set)] The countability of the set W of all the words of a finite alphabet is proved by other means than the usual countability proof. The latter consists in constructing a bijection with N, i.e., in constructing a list. But it is impossible to construct a list of all words. It is impossible to construct a bijection between W and N. Why should it be possible to construct a bijection between N and N? > > > > > Note however that in order to attach a definite cardinal number to > > every set, well-ordering must be possible for every set. That is why > > Cantor insisted on well-ordering of all sets. > > Piffle. Well-ordering has nothing to do with the existence or not of > at least one bijection. > You are wrong. If a set cannot be well-ordered, it cannot be assigned a definite cardinal number. In order to assign a cardinal, at least one bijection to an ordinal is required. This requires at least one well-ordering. Regards, wM
From: mueckenh on 21 Nov 2006 16:07 Franziska Neugebauer schrieb: > > Do you want to posit, that there is some line identical to the diagonal? > That is not (yet) proven. The number of elements of the diagonal is larger than all n: E omega A n : omega > n. The lines have only finitely many elements. In a linear order of finitely many elements n exactly the following is implied: If for every n there exists an line L(n) such that L(n) contains all elements m <=n then there exists a line L such that for every n, L contains all elements m <=n . This is valid for the finitely many elements of every line. (How many lines there are is irrelevant.) All elements of the diagonal are elements of the lines UL(n). Therefore all elements of the diagonal are elements of a finite line. Therefore: "E omega A n : omega > n" is false. Regards, WM
From: mueckenh on 21 Nov 2006 16:09 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > Franziska Neugebauer schrieb: > >> >> > 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. > >> > > >> > You wrote: "The cardinality of omega is |omega| not omega." > >> > >> And what I meant was: "The cardinality of X is |X| not X". I have > >> already clarified that my wording was misleading. > > > > You tried to say "misleading", but your wording was wrong. > > I will not rectify this once again. > > > The cardinality of omega is omega as well as |omega|. > > Under the common definition the cardinality of omega, also written as > |omega|, is omega. > > Anyhow. I still see no support for your pretended interpretation (****), > which I did not meant to write and did not write in that form. > > >> > Shall this sentence of yours express a difference between |omega| > >> > and omega or not? > >> > >> What exactly is so hard to understand? The cardinality of a set X is > >> (written) |X| and -- in the case of omega -- equals under the common > >> definition to omega. > > > > Therefore, the sentence "The cardinality of omega is |omega| not > > omega" is false? Or is it not false? > > Perhaps D. Marcus' understanding in > <MPG.1fcbb0f0d873ccdd989977(a)news.rcn.com> > may help you to cope with my sentence under discussion. > You wrote: 1) "The cardinality of omega is |omega| not omega." And after learning from me that this is wrong, 2) "The cardinality of omega, also written as |omega|, is omega". Now I am interested whether or not this is a contradiction in your eyes. If you say, no, it is slightly misleading but it is not a contradiction, then I am absolutely clear that ZFC will remain free of contradictions forever. But then I can tell the young students who not yet worship ZFC one more example of the logic of the worshepherds of transfinity. Regards, WM
From: Virgil on 21 Nov 2006 16:10
In article <1164106359.974111.298620(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > David Marcus wrote: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > [...] > > >> You wrote: "The cardinality of omega is |omega| not omega." > > >> > > >> Shall this sentence of yours express a difference between |omega| > > >> and omega or not? (Now I recognize why it is so difficult to convince > > >> the proponents sof set theory.) > > > > > > Franziska explained that what [s]he meant was that the notation for > > > the "cardinality of omega" is "|omega|", not "omega". It turns out > > > (using a standard definition for cardinality) that |omega| = omega. > > > > Thanks for the confirmation of understandability. > > A very sensible and understanding human being which not even can > distinguish between female and male names. In English, there are a large number of given names which are given to either gender, so that one cannot always be sure. For example, there someone who posts to this Ng as "[Mr.} Lynn ....." because "Lynn" is one of those androgenous names. To someone not familiar with naming habits in languages foreign to them, it is an understandable error. Can WM guarantee to get the correct gender for given names of, say , Finnish, or Japanese, posters? > > But the same gap of understanding becomes visible in his understanding > of technical terms. It turns out that also the *notation* for the > "cardinality of omega" is "omega". Not always. This would require that one use a specific definition of cardinality which is not universal. > > Regards, WM |