An uncountable countable set
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From: mueckenh on 25 Jul 2006 07:22 Dik T. Winter schrieb: > > In set theory "infinity" means "actual infinity". Potential infinity > > can neither be counted nor be surpassed. > > Read Cantor where he wrote that w is not actual infinity but potential > infinity, because you could not count to it. About the surpassing I am > not so sure. > I would have read it if he had written such a nonsense. But he did not. Cantor, Werke, p. 195, footnote: Das Zeichen oo, welches ich in Nr. 2 dieses Aufsatzes [hier S. 147] gebraucht habe ersetze ich von nun an durch w, weil das Zeichen oo schon vielfach zur Bezeichnung von unbestimmten [d. h. potentiellen] Unendlichkeiten verwandt wird. For people like you he said: Cantor, Werke, p. 374: "Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende Größe, letztere ein in sich festes, konstantes, jedoch jenseits aller endlichen Größen liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das eine mit dem andern verwechselt wird." > > > > The proof is given by the list: > > > > > > > > 1 0.1 > > > > 2 0.11 > > > > 3 0.111 > > > > ... ... > > > > w 0.111... > > > > > > That is neither a proof, nor a list. > > > > The list is given in the upper part. Call the whole an extended list. > > It shows that w is not the number of the naturals, because w appears in > > the sequence of ordinal numbers only once. > > That is only a statement, *not* a proof. Why is the ordinality of the left column of the extended list another than the ordinality of the right column? Both columns give exactly *the same* numbers - only in different representations. This *is* a proof that there is no transfinite ordinality at all, neither w nor w + 1. Regards, WM
From: mueckenh on 25 Jul 2006 07:24 David R Tribble schrieb: > Dik T. Winter wrote: > Mueckenh writes: > >> But it is obvious for *all* *finite* *unary* number. And only > >> finite unary numbers are involved in the second list below: > >> > >> The *- sum of > >> 0.1 > >> 0.11 > >> 0.111 > >> ... > >> 0.111... > >> > >> 0.111... is not a finite unary number. Rather, I would say it is not a > >> unary number at all. > >> 0.111.. is not a terminating rational number. > >> [...] > > > > The list: > 0.1 > 0.11 > 0.111 > ... > contains only terminating rational numbers (fractions), all of them of > the form: > f(n) = sum{i=1 to n} 10^-i, for all n=1,2,3,... > > This means that every f(i) contains a finite number (n) of 1 digits, > and likewise so does f(i+1) that follows f(i). So we conclude that > the list is composed entirely of finite terminating fractions. > > We also conclude, for exactly the same reasons, that the number > 0.111111... > is not in the list, because it is not a terminating fraction, and > because it is not of the form: > f(n) = sum{i=1 to n} 10^-i We know that every digit which can be indexed by a natural number is in the list. If 0.111... is not in the list, then not every digit can be indexed. This is simple logic: (every digit of n can be indexed) ==> (n is in the list) not (n is in the list) ==> not (every digit of n can be indexed) Regards, WM
From: Franziska Neugebauer on 25 Jul 2006 08:06 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > In set theory. It is full of the actual infinite. >> >> When I ask for "usage of terms" I want you to name contemporary >> papers/books on set thery that mention/use these terms. > > 1976 is too old? Preface, Wolfgang. You are refering to an elucidation in its preface on page 6. How many times does the book mention and use those two terms on the remaing pages? F. N. -- xyz
From: mueckenh on 25 Jul 2006 08:08 Dik T. Winter schrieb: > > > In my opinion the > > > only sane way to define the *+ sum of all elements is to define it as > > > lim{n -> oo} An > > > but you apparently disagree. > > > > That is only a convention. We do not know if it holds. In order to find > > out, my list was designed. Therefore we cannot use it as input. > > What do you mean here? We do not know if it holds when it is not defined. If a column contains at least one 1, then its *+sum is 1. Otherwise it is 0. This is a clear definition. > > I do not see why it cannot be correct. If you have a finite list, the *+ > sum is the last element of the list. If you have an infinite list, there > is no last element, so the *+ sum can not be the last element. And indeed, > even the assumption that the *+ sum of the list (say K) is one of the > elements (say An) leads to a contradiction. Because K *+ A_(n+1) is not > equal to K but equal to A_(n+1), so the assumption is false. You are > continuously ignoring that (*+){i = 1 .. oo} Ai is an ongoing process that > will never be complete, if we do the individual *+ sums. You *can not* > count to w. I need not count to w. If a column contains one 1 then the *+sum is 1. This is a definition like that of Cantor's list. > > > > Except that there is no digit position with ordinal number w in the > > > standard decimal notation. > > > > Yes, just that is the problem. All numbers of the true list have a > > finite number of digits. If it is claimed that all digits of 0.111... > > can be indexed, then 0.111... has omega 1's and then it belongs to the > > position omega and then omega is necessary to index it. > > The last conclusion requires proof (and it is false). What is the case > is that for every Ai the indexing of digits will terminate at some point, > and the indexing of digits of 0.111... will never terminate when we start > at the first digit and go on. Because there will always be a further > digit. (As there will always be a further An.) But each one will be finite. (every digit of n can be indexed) ==> (n is in the list) not (n is in the list) ==> not (every digit of n can be indexed) > > > > What is the ordinal number of its last digit (and please mind > > > that in decimal notation the ordinal numbers of the digits are natural > > > numbers)? > > > > The last digit of 0.111... has no ordinal number. > > Indeed, but the reason is that there is no last digit. w is still a > potential infinity, you can not count to it (for one thing, it has > no predecessor). Why does the left column of my extended list differ from the right one? There are the same numbers given, only their representation is different. > > > Therefore the set of > > its 1's and the set N has no ordinal number. Therefore it is wrong to > > define omega as the LUB of the naturals *and simultaneously as their > > cardinality*. > > Eh? The ordinal number of the ordered set N (in the natural order) is w, > the cardinal number of N is aleph-0 (as is the cardinal number of all > countable sets). The least ordinal of a class is also its cardinal number. If you say N has the cardinal number omega you are not wrong today. Even Cantor did so at his later times. Regards, WM
From: mueckenh on 25 Jul 2006 08:15
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> > In set theory. It is full of the actual infinite. > >> > >> When I ask for "usage of terms" I want you to name contemporary > >> papers/books on set thery that mention/use these terms. > > > > 1976 is too old? > > Preface, Wolfgang. You are refering to an elucidation in its preface on > page 6. How many times does the book mention and use those two terms on > the remaing pages? I am not in possession of the book but have only my notes. But he frequencay of mentionings may be high though unimporant. The meaning of "infinity" in set theory is clearly the actual one. Every expert knows this. Compare Jech's statement. Or compare Hilbert or Poincaré. That is just the clue, and it has not chaned since Cantor's times. Hilbert (1926, S. 167). "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." But here is some newer author: Lorenzen: Die endlichen Weltmodelle der gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft eines Gedankens einer aktualen Unendlichkeit mit der klassischen (neuzeitlichen) Physik zu Ende gegangen ist. Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual-Unendlichen in die Mathematik, die explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor begann. The strange thing is not the actual infinite but the fact that everyone except you knows the facts and requires no discussion about that point. Regards, WM |