Cantor Confusion
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From: Franziska Neugebauer on 9 Jan 2007 10:02 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> L_D does not exist >> >> so the union of line ends including the end of L_D does not exist. >> > >> > Therefre a complete, finished diagonal does not exist. >> >> As far as I can see is the diagonal "only potentially infinite". >> Nobody claimed that it is "complete". Nonetheless, the conclusion >> that a single line L_D does not exist remains valid. > > If the diagonal is not complete, then it has a largest element. No. I do refer to "complete" in the sense of "finished" in contrast to "potential" inifite. Since neither the list and hence nor the diagonal is finite there are no largest elements. OTOH since we do not assume every member of the list or the diagonal to provably exist, the list and the diagonal are "potential" infinite. This said William Hughes has shown that the assumption of the existance of a "potentially infinite" "last line" L_D leads to a contradiction. Hence "L_D exists" is wrong. > A line containing this element (and all smaller ones) does exist. Since there is no largest element in "potentially" infinite sets (in "actual/complete/finished", too) this sentence makes no sence at all. > If it turns out, that the diaogonal has a larger element, then it > turns out that a line containing this (and all smaller ones) does also > exist. > In any case, a line containig all elements of the diagonal does exist. Proof? >> >> > Fine. There are no infinite sets. There is only potential >> >> > infinity. Every line including the diagonal >> >> >> >> The diagonal is not a line. The diagonal is a potentially >> >> infinite set. >> > >> > Correct. Therefore it does not contain all natural numbers (as >> > indexes) because then it would be actually infinite. >> >> Please pay attention to your wording! The diagonal _contains_ numbers >> (values). Not "as indexes", but as value of the diagonal _at_ a >> certain index. So your statement is rather meaningless. > > To define what you are talking about: The diagonal of the EIT contains > only digits 1. > > 1 > 11 > 111 > ... > But these digits can be indexded by natural numbers 1,2,3,... > > The diagonal therefore, contains natural numbers as indexes as I said. Can you define "contain as indexes" mathematically? F. N. -- xyz
From: Franziska Neugebauer on 9 Jan 2007 10:22 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> > The finite union of two or more finite trees is a finite tree. >> >> > An infinite union of finite trees is the infinite tree. >> >> >> >> What is an infinite union? (Please give a definition) >> > >> > An infinite union of trees is the union of all (or nearly all) >> > finite trees which reach from level 0 to level n where n eps N. >> >> 1. Please give a definition of "union of all finite trees". > > Definitin: Denote the nodes of the tree by > > (0,0) > (1,0) (1,1) > (2,0a) (2,1a) (2,0b) (2,1b) > ... > (n,0a) (n,1a) ... > > The union of all trees up to the n-levels tree is > > {(0,0)} U {(1,0), (1,1)} U .. U {(n,0a) (n,1a) ...} > > Example: The union of the one-level tree and the two-levels tree is > > {(0,0), (1,0), (1,1)} U {(0,0), (1,0), (1,1), (2,0a), (2,1a), (2,0b), > (2,1b)} 1. "Definition By Example" is considered Bad Pratice. 2. The union operators are not "evaluated". How do they evaluate? 3. Why don't you take the standard graph theoretical approach? (Hint: graph G = (V, E), V: set of vertices, E: set of edges). 4. What is "infinite" in "infinite union of trees which reach from level 0 to level n e N"? I can only spot finitely many trees having finitely many levels. >> 2. Please give a definition of "union nearly all finite trees". > > Union of all trees with a finite number of exceptions. Of what? Please define exception. F. N. -- xyz
From: Virgil on 9 Jan 2007 12:53 In article <1168347129.804175.135070(a)38g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > A: L_D is a line, therefore L_D has a largest > > > > element. > > > > > > > > B: L_D contains any element that can be shown to exist > > > > in the diagonal, therefore L_D does not have a largest > > > > element. > > > > > > > > Contradiction. Therefore L_D does not exist. > > > > > > Therefore the union of line ends including the end of L_D does not > > > exist. > > > > The union of line ends is not a line. > > But it is supposed to exists as the complete, finished diagonal. That does not make it a line, but does make it a "diagonal". > > > L_D does not exist > > so the union of line ends including the end of L_D does not exist. > > Therefre a complete, finished diagonal does not exist. Sure it does. It just does not contain the end of nonexistent lines like L_D. > > > > > > > Fine. There are no infinite sets. There is only potential > > > infinity. Every line including the diagonal > > > > The diagonal is not a line. The diagonal is a potentially infinite > > set. > > Correct. Therefore it does not contain all natural numbers (as indexes) > because then it would be actually infinite. It is actually infinite in ZFC. > > > A line is not a potentially infinite set > > Correct. A line is or is not. But the system of lines is potentially > infinite. In ZFC it is actually infinite. > > > > > has a greatest element > > > until the existence of a greater one is shown. > > > > However, and contrary to your repeated claim, there is no > > single line which contains every element that can be shown > > to be in the diagonal. > > Every element that can be shown to be in the diagonal does exist as the > end of a line. Yes? > But not every end of a line does belong to a line? That is a surprising > aspect of existence. In ZFC, for every natural there is a larger natural but no natural that is larger than every other natural. Thus for every natural there is a "line" ending with it, but no "line" containing every natural, and the "diagonal" is the endless set of all naturals. > By showing that an element of the diagonal exists, you show that the > line contaning it and every smaller element exists. True, but irrelevant.
From: Virgil on 9 Jan 2007 12:57 > > The diagonal, by not having a last member, is not a line, all of which, > > by definition, must have last members. > > But the diagonal consists of last members. An infinite set can consist of finite members. So what? > If there was no line for > each member of the diagonal, which included this and every smaller > member, then the diagonal would not exist. In ZFC or NBG, the finite ordinals are lines and the first limit ordinal is the diagonal. This model disproves your thesis. > Therefore, the diagonal > contains only such elements with indexes {1,2,3,...n} which are > simultaneously in a line - in one line. All together! In ZFC or NBG, the finite ordinals are lines and the first limit ordinal is the diagonal. This model disproves your thesis.
From: Virgil on 9 Jan 2007 13:00
In article <1168348382.519894.3800(a)51g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > That is merely the embedding of the smaller into the larger, it is not a > > union. > > Embedding is a union. > The union given above is simply the union > {(0,0), (1,0), (1,1)} U {(0,0), (1,0), (1,1),(2,0a), (2,1a), (2,0b), > (2,1b)} How does WM embed these? A D / \ / \ C D E F |