Cantor Confusion
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From: David Marcus on 21 Jan 2007 12:59 Carsten Schultz wrote: > mueckenh(a)rz.fh-augsburg.de schrieb: > > It is interesting: > > William knows that T1 and T2 exist, but T1 is different. > > Virgil is also accepting T1 but, like William, sees only finite trees > > therein. > > Dik knows that T1 contains infinite trees, but doubts that its paths > > are the union of the paths of the finite trees. > > You doubt the existence of T1 (seeing that otherwise set theory is > > contradiced?). > > > > Couldn't you get to a consensus? It would spare me a lot of work. > > Since your writings are so confused and you never give exact definitions > or arguments, people have to guess what you mean. Sometimes they guess > differently. It would save everybody a lot of time if you would be > precise in your statements. However, you do not seem to be capable of > this. Also, the errors in your arguments would be even more obvious, > maybe even to you, if your arguments were stated precisely. The errors would be even more obvious (and the thread more interesting/amusing to read) if the people who reply to WM repeat important definitions. I can't keep all the different definitions straight! -- David Marcus
From: G. Frege on 21 Jan 2007 13:24 On Sun, 21 Jan 2007 18:55:37 +0100, Carsten Schultz <carsten(a)codimi.de> wrote: > > It would save everybody a lot of time if you would be precise in > your statements. However, you do not seem to be capable of this. > > Also, the errors in your arguments would be even more obvious, > maybe even to you, if your arguments were stated precisely. > I doubt that WM is interested in any errors in his own argumentation. He's only seeking for errors ("contradictions") in modern set theory. Remember, if errors in his "arguments" are pointed out to him, he usually prefers to ignore that. F. -- E-mail: info<at>simple-line<dot>de
From: Franziska Neugebauer on 21 Jan 2007 13:42 David Marcus wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> It is interesting: >> William knows that T1 and T2 exist, but T1 is different. >> Virgil is also accepting T1 but, like William, sees only finite trees >> therein. >> Dik knows that T1 contains infinite trees, but doubts that its paths >> are the union of the paths of the finite trees. >> You doubt the existence of T1 (seeing that otherwise set theory is >> contradiced?). >> >> Couldn't you get to a consensus? It would spare me a lot of work. > > The problem is that people use different definitions. It would help if > each person would include their definitions of important objects > (e.g., T1) in each post. WM is the one who claims: http://de.wikipedia.org/w/index.php?title=Bin%C3%A4rbaum&diff=26486595&oldid=26457398 ,----[ wikipedia ] | === Darstellung der reellen Zahlen des Intervalls [0, 1] === | - Der unendliche bin�re Baum dient zur Bin�rdarstellung aller reellen | Zahlen aus dem Intervall [0, 1] in Form von Pfaden (Knotenfolgen). | | - Ein endlicher bin�rer Baum mit ''n'' Ebenen besitzt weniger Pfade | als Knoten. Bis zur Ebene ''n'' > 0 gibt es 2<sup>''n''</sup> Pfade | und 2<sup>''n''+1</sup> - 1 Knoten. | | - Die Vereinigung aller endlichen bin�ren B�ume ist bez�glich aller | Knoten identisch mit dem unendlichen bin�ren Baum. Nach Aussage der | transfiniten Mengenlehre besitzt der unendliche bin�re Baum jedoch | mehr Pfade als Knoten, also mehr Pfade als die Vereinigung aller | endlichen bin�ren B�ume. `---- My translation of the last paragraph: "The union of all finite binary trees is with regard to all nodes identical with the infinite binary tree. According to transfinite set theory the infinite binary tree posesses more paths than nodes, i. e. more paths than the union of all finite trees." My objection to WM is: He has not yet defined what "union of all finite binary trees" means. So to me his texts in wikipedia and in sci.math are more or less meaningless. Furthermore I have objected to his usage of the notation T(1) U T(2) ... (inf-u) The notation T(m) U T(n) means - according to his definition of finite union of finite trees - with m, n being the depths e N of two complete finite binary trees nothing else than T(m) U T(n) := T(max(m, n)). (fin-u) Application to (inf-u) it would imply T(1) U T(2) ... := T(max(N)) Since N has no maximum the "union" remains undefined. Even if we now use T(m) U T(n) := T(sup(m, n)). (fin-u') and and try to define (inf-u) by T(1) U T(2) ... := T(omega) it is still left open what T(omega) shall mean. One has to take special care for the notation: The union symbol "U" does not mean the usual set theoretical union. If it would even a simply union of two finite trees would not in general be a tree. Therefor we can not use the axiom of union to claim that the "union of all finite trees" exists _and_ is a tree. F. N. -- xyz
From: David Marcus on 21 Jan 2007 14:00 G. Frege wrote: > On Sun, 21 Jan 2007 18:55:37 +0100, Carsten Schultz > <carsten(a)codimi.de> wrote: > > > It would save everybody a lot of time if you would be precise in > > your statements. However, you do not seem to be capable of this. > > > > Also, the errors in your arguments would be even more obvious, > > maybe even to you, if your arguments were stated precisely. > > > I doubt that WM is interested in any errors in his own argumentation. > He's only seeking for errors ("contradictions") in modern set theory. > Remember, if errors in his "arguments" are pointed out to him, he > usually prefers to ignore that. His arguments can't be wrong because they are obvious! -- David Marcus
From: stephen on 21 Jan 2007 14:08
David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > G. Frege wrote: >> On Sun, 21 Jan 2007 18:55:37 +0100, Carsten Schultz >> <carsten(a)codimi.de> wrote: >> >> > It would save everybody a lot of time if you would be precise in >> > your statements. However, you do not seem to be capable of this. >> > >> > Also, the errors in your arguments would be even more obvious, >> > maybe even to you, if your arguments were stated precisely. >> > >> I doubt that WM is interested in any errors in his own argumentation. >> He's only seeking for errors ("contradictions") in modern set theory. >> Remember, if errors in his "arguments" are pointed out to him, he >> usually prefers to ignore that. > His arguments can't be wrong because they are obvious! And Han claims to understand them, which is the definition of mathematical truth. :) Stephen |