Cantor Confusion
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From: mueckenh on 21 Nov 2006 05:49 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? > > 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. 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. Regards, WM
From: mueckenh on 21 Nov 2006 05:52 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. 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". Regards, WM
From: mueckenh on 21 Nov 2006 05:54 Virgil schrieb: > Every element except the first is "outside" at least one line. > > While it is true that there is no diagonal element that is "outside" of > EVERY line, that is quite a different issue. > > No, just this is the decisive issue. > > In ZF, statements with counterexamples are not theorems. One counter example contradicts ZFC: There is not one single element of the diagonal which is not contained in a line. This line contains this and all preceding elements. Regards, WM
From: Rainer Willis on 21 Nov 2006 07:28 Heinz Mau schrieb: > Am Tue, 21 Nov 2006 09:24:54 +0100 schrieb Eckard Blumschein: > >> Ich schrieb "w?re" weil ich einen entweder ?bersehenen oder, was >> wahrscheinlicher ist, sorgf?ltig versteckten kategorischen Unterschied >> zwischen dem originalen Kontinuumsbegriff und dem sogenannten >> Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. > > You are only a stupid little girl from germany. > > Du hast nur gezeigt, wie dumm Du bist. > > Sach mal, an der Uni Sommerpause oder wieso kannst du auf einmal Deine > ganze Zeit wieder hier verplempern?
From: Rainer Willis on 21 Nov 2006 07:31
Rainer Willis schrieb: > Heinz Mau schrieb: >> Am Tue, 21 Nov 2006 09:24:54 +0100 schrieb Eckard Blumschein: >> >>> Ich schrieb "w?re" weil ich einen entweder ?bersehenen oder, was >>> wahrscheinlicher ist, sorgf?ltig versteckten kategorischen Unterschied >>> zwischen dem originalen Kontinuumsbegriff und dem sogenannten >>> Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. >> >> You are only a stupid little girl from germany. >> >> Du hast nur gezeigt, wie dumm Du bist. >> Sach mal, an der Uni Sommerpause oder wieso kannst du auf einmal Deine >> ganze Zeit wieder hier verplempern? Sorry, irgendwie schiefgelaufen, diese posting ... :-( Gru?, Rainer |