Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 11 Jan 2007 07:47 Franziska Neugebauer schrieb: > > i.e., they are always finite but have no upper bound and, therefore, > > no fixed last line. > > This makes no sense. For the following reasons: > > 1. Sets in general have no lines at all. But in this special case (IET) they have. > > 2. "Always finite" makes no sense or is contradictory. "Always" refers > to a temporal aspect which is not present in mathematics due to the lack > of time. Finite in size implies having a last element. This and your further questions are answerd by the following texts. Meine Begriffserfassung des Transfiniten schliesst eigentlich und ursprünglich den "Process", weil derselbe eine "Veränderung" bedeutet, aus; nach mir ist transfinit = bestimmt, grösser als jedes noch so grosse Endliche derselben Art, trotzdem aber einer Vermehrung noch fähig. Der letzteren Eigenschaft wegen, ist nun allerdings im Gebiete des Transfiniten selbst ein Veränderung denkbar, wie etwa aus der Ordnungszahl omega werden kann omega + 1, daraus omega + 2 u.s.w. Es wären also auch transfinite Processe denkbar, sofern man darunter Processe im Gebiete des Transfiniten versteht. Aber ein eigentlich transfiniter Process scheint nach meiner Auffassung des "Transfiniten" nicht möglich, weil hier die beiden einander ausschließenden Prädicate "bestimmt = constant" und "veränderlich" verbunden wären. [Georg Cantor in a letter to Harnack, Nov. 3,1886] Will man in Kürze die neue Auffassung des Unendlichen, der Cantor Eingang verschafft hat, charakterisieren, so könnte man wohl sagen: in der Analysis haben wir es nur mit dem Unendlichkleinen und dem Unendlichgroßen als Limesbegriff, als etwas Werdendem, Entstehendem, Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. Aber das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir z.B., wenn wir die Gesamtheit der Zahlen 1,2,3,4, ... selbst als eine fertige Einheit betrachten oder die Punkte einer Strecke als eine Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des Unendlichen wird als aktual unendlich bezeichnet. [David Hilbert: Über das Unendliche, Math. Ann. 95 (1925) p. 167] Regards, WM
From: mueckenh on 11 Jan 2007 07:52 Franziska Neugebauer schrieb: > >> > A potentially infinite quantity (set or not) is always finite. > >> > >> There is no time in maths. > > > > So you write these letters and develop new ideas in zero time? > > My posts and my ideas do not take place *in* maths. They take place in > the real _physical_ world. So does the contents of your posts and ideas, in part mathematics. > > > There is no existence outside of time. > > _Mathematical_ existence is not to be confused with physical existence. > A mathematical entity x exists if there is a proof of "x exists". There is no proof existing outside space and time. > > As I have pointed out many times before: If and if so how the physical > world determines our reasoning is off topic in sci.math. Please consult > the neuro sciences groups for that complex of issues. > > This said your statement "A potentially infinite quantity (set or not) > is always finite" makes no sense. Read the texts by Cantor and Hilbert given in my recent contribution. > >> > Therefore in a linearly ordered set here is a last element. > >> > Contrary to the claim of set theorists, a set is not fixed in > >> > reality. > >> > >> When your arguments (by the way: What exactly are your arguments?) > > > > Here you can read it: > > > > Theorem. The set of real numbers in [0, 1] is countable. > > Please create a new thread with an appropriate topic. If you won't do, I > will help you. No. You asked for my argumenrts. I answered. That's enough. Regards, WM
From: Franziska Neugebauer on 11 Jan 2007 08:00 mueckenh(a)rz.fh-augsburg.de wrote: ,----[ <45a62aa2$0$97272$892e7fe2(a)authen.yellow.readfreenews.net> ] | > He misundertands the meaning of potential infinity. | | 1. Please explain what exactly he misunderstands. | | 2. Does he use a wrong definition of "potentially infinite set"? | If so, please give the "right" one, you want to base your game on. `---- > Franziska Neugebauer schrieb: >> > i.e., they are always finite but have no upper bound and, >> > therefore, no fixed last line. >> >> This makes no sense. For the following reasons: >> >> 1. Sets in general have no lines at all. > > But in this special case (IET) they have. What we have been discussing was: ,----[ <45a62aa2$0$97272$892e7fe2(a)authen.yellow.readfreenews.net> ] | >> > Don't you know that? Have you another definition of potential | >> > infinity? Do you think it is actual infinity? | >> | >> The question is whether you accept (for the sake of reasoning) that | >> there are "potentially infinite" sets. I. e. sets which are not | >> finite but also not provably-infinite in the sense of "actual | >> inifinite". | > | > i.e., they are always finite but have no upper bound and, therefore, | > no fixed last line. | | This makes no sense. For the following reasons: | | 1. Sets in general have no lines at all. `---- >> 2. "Always finite" makes no sense or is contradictory. "Always" >> refers to a temporal aspect which is not present in mathematics due >> to the lack of time. Finite in size implies having a last element. > > This and your further questions are answerd by the following texts. I did not pose any questions. I have informed you about the fact that there is no time and hence no temporal process _in_ math. F. N. -- xyz
From: Franziska Neugebauer on 11 Jan 2007 08:21 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> > A potentially infinite quantity (set or not) is always finite. >> >> >> >> There is no time in maths. >> > >> > So you write these letters and develop new ideas in zero time? >> >> My posts and my ideas do not take place *in* maths. They take place >> in the real _physical_ world. > > So does the contents of your posts and ideas, in part mathematics. This does not invalidate the fact that in mathematics there is no notion of time. And hence in mathematics there are no temporal processes. This said "always" is not a intra-mathematically defined notion. From this follows: "A potentially infinite quantity (set or not) is always finite" is a meaningless sentence. >> > There is no existence outside of time. >> >> _Mathematical_ existence is not to be confused with physical >> existence. A mathematical entity x exists if there is a proof of "x >> exists". > > There is no proof existing outside space and time. This is an argument supporting which claim? >> As I have pointed out many times before: If and if so how the >> physical world determines our reasoning is off topic in sci.math. >> Please consult the neuro sciences groups for that complex of issues. >> >> This said your statement "A potentially infinite quantity (set or >> not) is always finite" makes no sense. > > Read the texts by Cantor and Hilbert given in my recent contribution. I don't understand the fragments you have posted. They represent obviously merely a philophical discussion which do cannot understand. Perhaps you better reword the texts in your own lingo. If you want to state a different opinion to Cantor/Hilbert you should do so, too. Hence your sentence "A potentially infinite quantity (set or not) is always finite" still makes no sense. If it makes any sense you could explain that sense in your own words. >> >> > Therefore in a linearly ordered set here is a last element. >> >> > Contrary to the claim of set theorists, a set is not fixed in >> >> > reality. >> >> >> >> When your arguments (by the way: What exactly are your arguments?) >> > >> > Here you can read it: >> > >> > Theorem. The set of real numbers in [0, 1] is countable. >> >> Please create a new thread with an appropriate topic. If you won't >> do, I will help you. > > No. You asked for my argumenrts. I answered. That's enough. Enough is enough. F. N. -- xyz
From: Franziska Neugebauer on 11 Jan 2007 08:55
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: [...] > Conclusion: > Every finite binary tree contains a finite set of path. True. > The countable union of finite sets is countable. True. The union of countably many countable sets is countable. > The set of paths is countable. 1. The set of all rooted _finite_ paths in an infinite tree is countable. The union of all finite paths is exactly this set of all rooted finite paths. 2. To "unary represent" every real in [0, 1] you also need _infinite_ paths. (i.e. 1/3 is not represented as finite path and hence not by any path in your union). > The set of real numbers in [0, 1] is countable. Non sequitur. F. N. -- xyz |