Cantor Confusion
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From: Virgil on 13 Nov 2006 19:15 In article <1163454806.449277.206770(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1163428970.972473.215570(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > Every column has a fixed number omega of terms. You can add 1 further > > > > > term and you will get omega + 1 terms in every column. Every line has > > > > > less than omega terms. If you add 1 term to every line, the number of > > > > > terms in every line remains less than omega. What about the diagonal? > > > > > It has to satisfy both conditions, which is impossible. > > > > > > > > No. Adding one element to each line does not change > > > > the supremum of the lengths of the lines, so it does > > > > not change the length of the diagonal. > > > > > > > > The number of terms in each line, n, is less than omega. > > > > The number of columns is the supremum of the number of terms > > > > in each line. The number of columns is omega. > > > > The number of lines is omega. > > > > The number of columns is equal to the number of lines. > > > > > > > > > > > > Now add one term to each line. > > > > > > > > The number of terms in each line, n+1, is less than omega. > > > > The number of columns is the supremum of the number of terms > > > > in each line. The number of columns is omega. > > > > The number of lines is omega. > > > > The number of columns is equal to the number of lines. > > > > > > PS: You forgot to consider the case that one term is added to each > > > column. The ordinal of the columns is then larger than omega. > > > > But the cardinal is not. Which is what you get. > > If bijecting ordered sets, I am interested in the ordinal number. When counting digits one needs to consider cardinality, regardless of any orderings. > Without order no cardinal. Did you forget why Cantor strived to order > any set? But different infinite ordinals have the same cardinality, and it is cardinality which is Cantor's "size" of sets. So claiming that different ordinality requires different cardinality, at least in this case, is just plain wrong! In fact, since the reals cannot be explicitly well ordered, we have no practical commonplace examples of ordinals having cardinality greater than that of omega.
From: Dik T. Winter on 13 Nov 2006 19:28 In article <1163428158.317887.311810(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > Cantor considered well-ordering as a first principle, > > > Zermelo introduced it at a first principle =3D axiom. Cantor was wrong, > > > Zermelo was right? > > > > Cantor did state it without suggesting either that it was a first > > principle or something else. He just assumed it. And he was wrong > > with that assumption. > > You are wrong. "Der Begriff der wohlgeordneten Menge weist sich als > fundamental f?r die ganze Mannigfaltigkeitslehre aus. Da? es immer > m?glich ist, jede wohldefinierte Menge in die Form einer > wohlgeordneten Menge zu bringen, auf dieses, wie mir scheint, > grundlegende und folgenreiche, durch seine Allgemeing?ltigkeit > besonders merkw?rdige Denkgesetz werde ich in einer sp?teren > Abhandlung zur?ckkommen." (Cantor, Collected works, p.169) Ah, I missed that one. So he uses it as an axiom. > > No. The above statement is not implied by the letter of Cantor to > > Mittag-Leffler. It is explicitly stated elsewhere. And it is wrong. > > To "count" a set of ordinality omega you do not need omega. In most > > cases you need omega, but there is one exception. Can you find it? > > To "count" a set of ordinality a with ordinals you need only the > > ordinals smaller than a. (And note that in counting with ordinals > > you start at 0, because that is the first ordinal.) > > So you think that counting all natural numbers does not require an > infinite number? We agree. Indeed. However, when the question is: "how many natural numbers are there", we need an "infinite number". Or when the question is, what is the ordinal number of that set, we need an "infinite number". -- 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 13 Nov 2006 19:37 In article <1163433510.518520.69770(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Well-ordering is so easy connected with AC that we can > > > state: Well-ordering has been assumed. > > > > Not in set theory. AC is an axiom that may or may not hold, depending > > on the branch you are following. > > Zermelo considered it as "has to be taken". Perhaps he did. In modern mathematics there is no such imperative. Euclid considered that the parallel postulate "has to be taken". This does not mean that in modern mathematics it has to be taken. It surprises me that in one case you consider the overthrowing of an axiom that was basic for centuries as valid, while in the other case you consider overthrowing an axiom that has been valid for a few decades as not valid. > > They were in fact axioms, although he did not state it as such. And the > > first principles he did chose where of course arbitrary. > > Oh, he would rotate in his grave if he heard you. Of course he only > assumed those first principles which were true in nature or reality. What is true about 3 in nature or reality? What is true about a set of numbers in nature or reality? > > Oh, well, do set theory without AC. No problem. There is a number of > > people doing it. > > Zermelo did not belong to this group. Yes, and so what? Euclid did not belong to the group that does either elliptic or hyperbolic geometry. -- 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 13 Nov 2006 19:37 In article <1163433653.585487.297660(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1163426814.926776.54020(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > May be so with some sets. A set of natural numbers includes its > > > cardinal number. > > > > I did not know that. Care to prove that the set {1, 3, 5, 7} includes > > the cardinal 4? > > Thank you for your attention. Correction: > An initial segment of natural numbers includes its cardinal number. Further correction. A finite initial segment of natural numbers includes its cardinal number. -- 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 13 Nov 2006 19:53
In article <1163453584.773656.46060(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1163353613.745777.63980(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: .... > > > I hoped to settle this question with my quote. > > > > How can your quote settle whether it was explicitly stated or not in my > > quote? > > Meanwhile I checked your quotes and could not find any hint that Cantor > explicitly or implicitly stated that a set of cardinality aleph_0 has > an omega-th element. See <J7E0uw.32v(a)cwi.nl>. "... sets of the first cardinality can be counted only through (with the aid of) numbers of the second class..." (Abhandlungen, p 213.) And that was the quote you denied he explicitly stated. He does not state (and I did not write that he did state) that a set of cardinality aleph_0 has an omega-th element, but that can easily be deduced from the above quote. > > > That did he mean with countable by numbers of class II. > > > You see, he did not use an omegath element. > > > > Yes, so what? > > Why then do you assert he said that every set with cardinality aleph-0 > had an omega-th element? That is *not* what I asserted. See above. I also assert that from the quote above it is easily deduced that there is an omega-th element. But in the above quote Cantor states a falsehood. > > > > That ordered set has ordinality 2*omega + 1. What is the problem? > > > > > > That each omega can be substituted by 1,2,3,... or say a,b,c,... > > > yielding 4 omega. > > > > You are again confusing a set with its contents. > > A set is its contents - and nothing more! There are no ghosts in > mathematics. Eh? > > If you want to look > > at ordinals as sets you can substitute omega with {1,2,3,...} or > > {a,b,c,...}, not with what you write. And in the sequence above they > > still remain a single element. > > There is a difference with Cantor' s notation. But that is outdated. Sorry, I do not understand what you write here. The set {{1, 2}, {3, 4}} has two elements. It's cardinality is 2. And each of the elements is a set containing two elements. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |