Cantor Confusion
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From: Franziska Neugebauer on 14 Nov 2006 06:56 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: >> >> >> > Franziska Neugebauer schrieb: >> >> >> [...] >> >> >> >> Are there really three vertices in WM's "triangle"? >> >> >> > If finished infinities [...] >> >> >> Verbiage. >> >> > Yes. But, sorry to see, it is the fundament of modern >> >> > mathematics. >> >> "Finished infinities" is your wording. >> > Precisely describing the fundament of modern mathematics. >> Cbjre Bs Oryvrs. > Always insisting on having the last word? There is no last word. F. N. -- xyz
From: mueckenh on 14 Nov 2006 07:07 Dik T. Winter schrieb: > 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, I don't. The parallel axiom is necessary, correct, true in any Euclidean plane. > > > 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? Cantor thought so, as I just quoted in my last letter to you. And I think so too. Therefore I ike him by far more than all the modern mathematicians. > > > > 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. But he was a bit suspicious, because he avoided the application of his 5th postulate as long as possible. I think that the rise of non-euclidean geometries was a big disadvantage of mathematics, because from then on every guy felt entitled to create his own system of axioms. Regards, WM
From: William Hughes on 14 Nov 2006 07:08 Franziska Neugebauer wrote: > 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: > >> >> >> > Franziska Neugebauer schrieb: > >> >> >> [...] > >> >> >> >> Are there really three vertices in WM's "triangle"? > >> >> >> > If finished infinities [...] > >> >> >> Verbiage. > >> >> > Yes. But, sorry to see, it is the fundament of modern > >> >> > mathematics. > >> >> "Finished infinities" is your wording. > >> > Precisely describing the fundament of modern mathematics. > >> Cbjre Bs Oryvrs. > > Always insisting on having the last word? > > There is no last word. > LOL
From: mueckenh on 14 Nov 2006 07:10 Dik T. Winter schrieb: > 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. Rejected. Regards, WM
From: mueckenh on 14 Nov 2006 07:26
Dik T. Winter schrieb: > 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.) In German: "während die Mengen erster Mächtigkeit nur durch (mit Hilfe von) Zahlen der zweiten Zahlenklasse abgezählt werden können" > 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. Cantor considered as "Anzahl" which can be determind by "abzaehlen" simply the ordinal number of a set. p. 213: Durch Umformung einer wohlgeordneten Menge wird, wie ich dies in Nr. 5 wegen seiner Wichtigkeit wiederholt hervorgehoben habe, nicht ihre Mächtigkeit geändert, wohl aber kann dadurch ihre Anzahl eine andere werden. How this "abzaehlen" is done I posted recently: 1,2,3,... = omega 2,3,4,...,1 = omega + 1 This does *not* mean that omega is an element of the set. That means, the Anzahl changes = the ordinal number changes. It cannot be deduced that Cantor thought that there was an omegath element. > > > > 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? {1,2,3} is the collection of, and a convenient expression to write that we are talking about, the numbers 1 ,2, and 3. > > > > 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. Yes, here you are talking about two unordered pairs of numbers. You use a convenient way to denote that. Nothing else stands behind the {, } symbols. In particular N = 1,2,3,... = {1,2,3,...}. That does not prevent to build a set {{1,2,3,...}, 1} with two elements. Regards, WM |