Cantor Confusion
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From: mueckenh on 10 Jan 2007 11:14 Dik T. Winter schrieb: > In article <1168351968.885524.129360(a)s34g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > No, that does AC (although the statement is indirectly). V=L is the axiom > > > of constructibility and states that every set is constructible. Most > > > mathematicians think it is false, but it can not be disproven and is > > > "consistent" with ZF. It *gives* a construction. > > > > But, alas, this construction cannot be communicated. It is and remains > > a top secret. Otherwise most mathematicians could easily be proved > > wrong by simply constructing a well-ordering of R. > > It can be communicated. Why does nobody dare to do so? Serious punishment by the high priests to be expected? > But you are not even willing to look at the > axiom and its consequences. I am not interested in consequences but in the *definition or construction* of a well-ordering of the reals. Regards, WM
From: Franziska Neugebauer on 10 Jan 2007 11:16 mueckenh(a)rz.fh-augsburg.de wrote: >> "Different line ends are always >> elements of one single line" is not true for the potentially >> infinite set of all "Different line ends". > > Of course it is. The potentially infinite set is always a finite set. First of all "potentially infinite" is an attribute not a noun. Since maths (with or without "potential infinite sets") does not entail a concept of time your last sentence does not make sence. As far as I can see William has proved that a "potential infinite" set is _not_ finite. > 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". >> > > L_D does not exist. >> > >> > Of course does it not exist, because the diagonal does not exist. >> > >> >> Make up your mind. Your repeated claim is that L_D does exist. >> > My claim was that L_D exists as a fixed line IF the complete diagonal > exists actually. (This was argued in order to disprove actual > infinity.) In the last say one hundered posts your have been discussing with William Hughes the issue under the presumption that the diagonal exists precluding the provable-existence of every of its elements. Since you now want to discuss the existence of L_D under the presumption that the diagonal "actually" exists, you are now discussing a different issue. Nonetheless L_D does not exist either (which is easily proved by reductio ad absurdum). > For the potentially infinite diagonal, a last line also exists, but > not as a fixed line. There is no time in maths. Hence your sentence is meaningless. Is it possible that you confuse a "potentially infinite" set with a computer program which step-by-step generates the set members? If so, than your arguing is obviously driven by wrong imaginations. William's proof that no last L_D exists even if the diagonal exists only "potentially" remains valid. If you want to attack his proof you must show an error _therin_. > Potenial infinity is changing (growing) finity. Says who? The pope? F. N. -- xyz
From: Franziska Neugebauer on 10 Jan 2007 11:30 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: ,----[ <45a3ae6b$0$97254$892e7fe2(a)authen.yellow.readfreenews.net> ] | 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 | exist[e]nce of a "potentially infinite" "last line" L_D leads to a | contradiction. Hence "L_D exists" is wrong. `---- So you agree to this part you have cut? Fine! >> Since there is no largest element in "potentially" infinite sets (in >> "actual/complete/finished", too) this sentence makes no sence at all. > > A potentially infinite quantity (set or not) is always finite. There is no time in maths. > 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?) are exhausted your physicalisms come up. > Avoiding this recognition maks present tansfinite set theory > absolutely useless for any science except itself. (It is like a bad > lantern which only enlightens its own ost.) Straw man. Political discussions about the worth of mathematics/set theroy as an auxilliary science to other sciences are of-topic. >> > 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? > > The existence of the diagonal (if existing) and the fact that a given > set of natural numbers is always a subset of a natural number. Why not say "the pope said so"? >> > 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? >> > That should not be necessary in this newsgroup. I did think so, too, until I met you. > The n-th digt carries the index n. Doesn't become the index n to heavy when n becomes too large? I mean nobody seems to care about the tiny poor digits. F. N. -- xyz
From: William Hughes on 10 Jan 2007 11:51 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > No. It is clear that given any two natural numbers, or indeed any > > > > > > set of natural numbers you can write down, there exists a single > > > > > > line which contains all the natural numbers you wrote down. But this > > > > > > line > > > > > > *can* depend on which set of natural numbers you write > > > > > > down. > > > > > > > > > > Of course. And if you write down all natural numbers, then you write > > > > > down L_D. > > > > > If the set of all natual numbers is complete, then L_D is complete. > > > > > If the set is potentially infinite, then L_D can change. It depends on > > > > > which set of natural numbers you just have. > > > > > > > > L_D cannot change. L_D is a line. A line cannot > > > > change. > > > > > > No. But the property of being the greatest line can and does change. > > > > Irrelevant. The question is whether L_D exists. > > It does. Does the tallest man exist? When did it start, when did it > cease? Recall: L_D is a line that contains every element that can be shown to be in the diagonal. The "tallest man" is something that can change. L_D is a line. A line cannot change. The analogy is not valid. > > > > > > > > > You are saying that for every maximum integer, n_m, > > > > that can be shown to exist, a line L_M(n_m) > > > > which contains all integers up to and including > > > > this maximum (i.e. all integers that have been shown to > > > > exist at this point) can be shown to exist. > > > > No one is disuputing this. However, you cannot find > > > > a single line L_D that contains every L_M(n_m) > > > > that can be shown to exist. > > > > > > Of course you cannot find that line. > > > > Your putative proof that assuming actual > > infinity leads to a contradiction is: > > > > It is possible to find L_D > > It is not assumed, but it is obvious that for every given set of > natural numbers there is one line containing it. But since the set of natural numbers can change this "one line" can change. It is not L_D. > > > > If you assume actual infinity then L_D > > does not exist. > > It does exist, in potential infinity. But it is not fixed. L_D is a line. A line is fixed. L_D does not exist. >That is a > property of potential infinity, because every set is finite. > > > > Therefore you cannot assume actual infinity. > > > > You now admit that it is not possible to find L_D, > > In actual infinity (everything including L_D being fixed) it is not > possile to find L_D. And it is also not possible to find L_D in potential infinity. > > > > Yes, if you assume actual infinity then the description > > of the lines L_M(n_m) changes slightly. However, these > > lines still exist. Assuming actual infinity does > > not lead to a contradiction. > > It is a contradiction. No. No line that can be shown to exist if you assume potential infinity ceases to exist if you assume actual infinity. There is no contradiction. - William Hughes
From: Franziska Neugebauer on 10 Jan 2007 11:57
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: >> >> > 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". >> > > ==================================================== >> > Definition: 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) ...} > > which is obviously the same as > > {(0,0)} U {(0,0), (1,0), (1,1)} U .. U {(0,0), (1,0), (1,1),..., > (n,0a) (n,1a) ...} > > End of definition. > ================================================= >> > >> > 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. > > The upper part contains the definition. To support a definition by an > exampe is good practice in most text books. I have asked for a definition of "union of all finite trees", not of "union of all finite trees upto some level" or of " >> 2. The union operators are not "evaluated". How do they evaluate? > > Read a book on set theory. I prefer graph theory. >> 3. Why don't you take the standard graph theoretical approach? >> (Hint: graph G = (V, E), V: set of vertices, E: set of edges). > > Because it veils the conradiction raising from the tree. So you confirm that a graph theoretical description of trees is sound whereas a Mueckenheimian description is inconsistent? Well done! >> 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. > > Then you think that there is no infinite union of naural numbers? 1. Do you agree that every n e N is finite? If so, what is "infinite" in "infinite union of trees which reach from level 0 to level n"? > Spot a natural number which is not, as a level, in the tree. In which tree? >> >> 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. > > In German: "Ausnahme". (It is usual to use "fast alle" in mathematics > of infinite sets. Just look it up in a book.) I see. You mean "union of nearly all finite trees" means "union of all finite but countably many"? So then please explain what the parentheses in your answer to my question mean. You wrote: ,----[ WM ] | 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. `---- What is the difference between a) union of all finite trees which reach from level 0 to level n where n eps N, and b) An infinite union of trees is the union of nearly all finite trees which reach from level 0 to level n where n eps N? F. N. -- xyz |