An uncountable countable set
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From: Dik T. Winter on 21 Aug 2006 08:13 In article <1156149533.333542.222290(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > But according to Cantor width 1 *is* reached while height 1 is not > > > reached, infinity in number does actually exist infinity in size does > > > not exist. > > > > I do not think so. What Cantor states is that a container exsits that > > has both width and height 1. > > False. According to Cantor the height is never 1, width is 1. I do not think so. With blocks of height 1/2^n and width 1 - 2^n, after k steps the total height is 1 - 2^k and the total width 1 - 2^k. When we complete we get the limiting case. But there is still no block at either height 1 or with width 1. And there is no smaller container that contains the complete stair. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 21 Aug 2006 08:21 In article <1156149770.217482.166370(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > > But there is no need to check each and every individual line. When I > > > > state: > > > > sum{i = 1 .. n} i = n * (n + 1) / 2 > > > > do I need to check for each and every n? > > > > The *definition* of the diagonal makes clear that it is different for > > > > each and every n. Like in the formula above I need not check for > > > > each individual n, I need not check for a difference for each and > > > > every n. .... > > Eh? It holds for every finite number but does not hold for an infinite > > number of finite numbers? What do you mean with that? If there are > > infinitely many finite numbers, I would say that as it holds for > > every finite number, it also holds for infinitely many finite numbers. > > What is the sum of infinitely many finite numbers? What is the sum of > all finite numbers? I state that it holds for infinitely many 'n'. Not that it holds for infinite 'n' (whatever that may be). > > > > 100 %? But with sets I would state that a set exists if all of its > > > > elements do exist and all of its subsets do exist. > > > > > > Is it possible? If ZFC is consistent, then it has a countable model. If > > > all of its subsets would exist, then it was uncountable. > > > > Is ZFC a set? > > The subsets of the set of the model are meant of course. But the number of subsets have no relation to the question of whether a model is countable or not. This is similar to stating: N is countable. If all of its subsets would exist, then it was uncountable. > > > > Oh. I must have missed something, because I have not seen a proof. > > > > Given an injection f: N -> P(N), why does the set > > > > M(f) = {n in N | n !in f(n)} > > > > not exist? > > > > > > Excuse me, the non-existing set is the triple {f, n, M_f(n)} > > > > Excuse me, Hessenberg was not talking about such triples. > > But I am doing so in order to show that his arguing concerns > impredicative definitions and is inconclusive.f is the mapping, n is a > natural number, M_f(n) is a set which contains all nongenerators, > including n if not including n which is mapped on M. Yes. Such triples do not exist. And that precisely shows why Hessenberg's proof was right. If there is a surjective mapping f from N to P(N) it is a requirement (of surjectivity) that such a triple *does* exist. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Albrecht on 21 Aug 2006 08:38 Tony Orlow schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > > > >> You have agreed to that this limit does not exist ("There is no L in > >> N"). So the sum is at least _not_ _finite_ (not in omega). > >> > > There is no L, nowhere. It is wrong to say that L is in |N and it is > > wrong to say that L is larger than any n e |N. Actual infinity does not > > exist! I told you that already several times. > > But refuse to understand what "to exist" means. In no case infinitely > > many differeces of 1 can exist unless infinitely many diferences of 1 > > do exist. But that means an infinite size. > > > > Regards, WM. > > > > Well, here you answer my question just posed to you in my last post. > There is no infinite number, in your opinion. But, how do you reconcile > this with the infinity of reals in any unit interval? If, between any > two distinguishable reals there is a third distinguishable from the > first two, then does this not imply that we can subdivide the unit > interval infinitely, yielding an infinite set of interval endpoints, aka > reals? Isn't the set of reals in (0,1] infinite, and how do you > characterize this number? > Hi Tony, how do you do? Maybe the following thoughts may help to find an answer to your question: A difference is, if we say: - the line (0,1] consists of infinite many points (reals) or - we can found infinite many points (reals) on the line. The first statement is wrong. A point has no extension. So, you can put as many points together as you want, you can't build up an extension as e.g. a line of lenght 1. We must conclude that we can only say: we can find infinite many points on a line. This is a potential infinity. Now the analogue phrases are: - the set of natural numbers consists of infinite many numbers or - we can found infinite many natural numbers. Here, the first statement is as wrong as above. So, the second statement must be the right one. The infinity of the amount of natural numbers is a potential one. An infinite set doesn't exist. Infinite sets are self contradicting. Greetings Albrecht S. Storz
From: Dik T. Winter on 21 Aug 2006 09:11 In article <J472w4.Lt1(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <1155885821.815144.187270(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > To be more correct. The first proof is not relying on the > > > representation of reals, the second (diagonal) proof is not > > > about reals at all. .... > > > The answer was: no. To clarify, in Cantor's diagonal proof > > > there are no dual representations, so there was no need for Cantor > > > to consider them. It is about infinite sequences of symbols. But > > > Mueckenheim was insidious there, as he answered no, while not giving > > > the clarification. > > > > Cantor considered his 2nd proof as being /valid/ for real numbers. With 2nd proof you mean the diagonal proof, I think? > > he did not use irrational numbers: "Aus dem in 2 Bewiesenen folgt > > nmlich ohne weiteres, da beispielsweise die Gesamtheit aller > > /reellen Zahlen/ eines beliebigen Intervalles sich nicht in der > > Reihenform w_1, w_2, w_3, ..., w_v, ... darstellen lt. > > Es lt sich aber von /jenem Satze/ ein viel einfacherer Beweis > > liefern, der unabhngig von der Betrachtung der Irrationalzahlen ist." > > (The italics are mine) > Ok, found and reformatted. I will give an English translation: From what has been proven in section 2 (*) follows that e.g. the set of real numbers in an arbitrary interval can not be put in a sequence w_1, w_2, w_3, ..., w_v, ... . It is however possible to construct a much simpler proof for that theorem (**) that is independent from the observation of irrational numbers." Now (*) means section 2 of the paper that shows the first proof. What does (**) mean? *Not* what you imply. It talks about the theorem stated in the first half of that paragraph, which I will give it first in German: In dem Aufsatze, betitelt: "ber eine Eigenschaft des Inbegriffs aller reellen algebraische Zahlen (Journ. Math. Bd. 77, S. 258) [hier S. 115], findet sich wohl zum ersten Male ein Beweis fr den Satz, da es unendliche Mannigfaltigkeiten gibt, die sich niet gegenseitig auf die Gesamtheit aller endlichen Zahlen 1, 2, 3, ...m v, ... beziehel lassen, oder, wie ich much auszudrcken pflege, die nicht die Mchtigkeit der Zahlenreihe 1, 2, 3, ..., v, ... haben. translated: In the article, titled: "ber eine Eigenschaft des Ombegriffs aller reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258) [hier S. 115], can for the first time be found a proof of the theorem that there are infinite sets that are not in bijection with the set of natural numbers 1, 2, 3, ..., v, ..., or, as I say in general, do not have the same cardinality as the natural numbers 1, 2, 3, ..., v, ... . So he does *not* consider the diagonal proof adequate for the reals. He considers it adequate and simpler than the first proof about the theorem that there are infinite sets with a cardinality not equal to the natural numbers. You were insidious again by omitting the first half of that paragraph. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: MoeBlee on 21 Aug 2006 13:11
Albrecht wrote: > We must conclude that we can only say: we can find infinite many points > on a line. This is a potential infinity. Do you have axioms for your mathematical theory of potential but not actual infinity? MoeBlee |