An uncountable countable set
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From: Richard Tobin on 23 Jun 2006 19:55 In article <e7hu56$9n2$1(a)nntp.aioe.org>, Dann Corbit <dcorbit(a)connx.com> wrote: >> Joshua 10:13 describes a miraculous event in which Joshua commands the >> sun to stand still, with the result that the day continues until the >> children of Israel had avenged themselves upon their enemies. This, >> it was argued, shows that the cycle of day and night is normally >> caused by the movement of the sun. >It shows no such thing. The sun moves in the sky (visually). Go and argue with some people who died hundreds of years ago. -- Richard
From: Dann Corbit on 23 Jun 2006 19:50 "Richard Tobin" <richard(a)cogsci.ed.ac.uk> wrote in message news:e7hv23$72f$1(a)pc-news.cogsci.ed.ac.uk... > In article <e7hu56$9n2$1(a)nntp.aioe.org>, Dann Corbit <dcorbit(a)connx.com> > wrote: > >>> Joshua 10:13 describes a miraculous event in which Joshua commands the >>> sun to stand still, with the result that the day continues until the >>> children of Israel had avenged themselves upon their enemies. This, >>> it was argued, shows that the cycle of day and night is normally >>> caused by the movement of the sun. > >>It shows no such thing. The sun moves in the sky (visually). > > Go and argue with some people who died hundreds of years ago. Dirt can't talk. They're dirt by now. That may have been the basis for the Catholic church's position (I don't know the history of that position very well), but if it was, it was a silly basis.
From: mueckenh on 24 Jun 2006 03:51 Dik T. Winter schrieb: > > This mapping is impossible. (See my article > > "On Cantor's Theorem", arXiv, math.GM/0505648, 30 May 2005.) > > Consider two cases: > 1. > a -> {} > 1 -> {1} > K = {} > so K is in the image > 2. > a -> {1} > 1 -> {} > K = {1} > so K is in the image. K is not in the image of *a natural number*. a is not a natural number! > > > 2) Consider a mapping |N --> P(|N) which need not be surjective but has > > to satisfy only one condition, namely that the set K = {k e |N & k /e > > f(k)} is in the image. This mapping is impossible. > > Right. > > > In both cases there is this impredicable request {f, k, K} which is > > impossible to satisfy. But in the proof by Hessenberg, you insist, it > > would prove non-surjectivity? > > Wrong. In the first case the condition can be satisfied, in the second > case it can not be satisfied. The request {f, k, K} cannot be satisfied in any case,. That is completely independent of surjectivity. > > Here is how we obtain it: > > Cantor said: A well-ordered set remains well-ordered, if finitely many > > or infinitely many transpositions are executed. Let's see what happens. > > Give a quote please. Where did he state that? G E O R G C A N T O R: GESAMMELTE ABHANDLUNGEN MATHEMATISCHEN UND PHILOSOPHISCHEN INHALTS Mit erläuternden Anmerkungen sowie mitmErgänzungen aus dem Briefwechsel Cantor - Dedekind Herausgegeben von ERNST ZERMELO Nebst einem Lebenslauf Cantors von ADOLF FRAENKEL1966 GEORG OLMS VERLAGSBUCHHANDLUNG HILDESHEIM p. 214: "Die Frage, durch welche Umformungen einer wohlgeordneten Menge ihre Anzahl geändert wird, durch welche nicht, läßt sich einfach so beantworten, daß diejenigen und nur diejenigen Umformungen die Anzahl ungeändert lassen, welche sich zurückführen lassen auf eine endliche oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je zweier Elemente." Cantor admitted infinitely many transpositions. I think he meant countably many. But more aren't needed. Regards, WM
From: mueckenh on 24 Jun 2006 03:54 Daryl McCullough schrieb: > > > Because well-orderings are not preserved by an infinite sequence > of transpositions. Why not? > > >Hence, the well ordered set of all positive rational does not exist. > > I just gave you one. Of course it exists. You gave a short finite sequence. Your *law* is not preserved in the infinite. > > When you prove something that you know to be false, then you have > to suspect something is wrong with your proof. In your case, you > know that there is a well-ordering of all the rationals, because > I just showed you one. So the question is: what is wrong with your > proof. The mistake is that well-orderings are not preserved by an > infinite sequence of transpositions. The mistake is that well-orderings are not preserved by an infinite sequence of numbers. Don't you see the contradiction? Why should an infinite sequence of numbers preserve well-order better than an infinite sequence of transpositions? --- If infinity existed. Regards, WM
From: mueckenh on 24 Jun 2006 03:55
Daryl McCullough schrieb: > mueckenh(a)rz.fh-augsburg.de says... > > >Cantor said: A well-ordered set remains well-ordered, if finitely many > >or infinitely many transpositions are executed. > > I don't believe that Cantor ever said that. But whether he did or not, > it's a false statement. As false as yours about a well-ordering of *all* rationals. Regards, WM |