Cantor Confusion
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From: Virgil on 14 Nov 2006 19:49 In article <1163504471.047770.83600(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1163433510.518520.69770(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > > > > > 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. > > > > The exercise will do him good. > > > After having visited you at midnight. > > > > >Of course he only > > > assumed those first principles which were true in nature or reality. > > > > Perhaps in WM's "reality" but WM's reality is quite different from > > everyone elses'. > > Not from Cantor's. He wrote, for instance to Killing, on April 5, 1895: > Was Herr Veronese dar?ber in seiner Schrift giebt, halte ich f?r > Phantastereien und was er gegen mich darin vorbringt, ist unbegr?ndet. > Ueber seine unendlich gro?en Zahlen sagt er, da? sie auf anderen > Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber > auf gar keinen Hypothesen sondern sind unmittelbar aus dem nat?rlichen > Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von > Willk?r, wie die endlichen ganzen Zahlen. > > Briefly: My infinite numbers are founded only on the natural notion of > sets. They are as necessary and free of arbitriness as the finite whole > numbers. What Cantor may have said in 1895 need not be binding in 2006. But in any case it does not support WM's delusions.
From: Virgil on 14 Nov 2006 19:51 In article <1163504837.420660.251800(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Lester Zick schrieb: > are. > > > > > >Try to construct as many numbers as you can using only 100 bits. Then > > >increase to 10^10 bits, then increase to 10^100 bits. More is not > > >available. I find this very convincing. > > > > I don't. It's a problematic argument at best. Based once again on a > > hypothetical finitude of the "physical" universe whatever that means. > > Here I cannot understand you. The accessible universe is finite, > allowing for not more than 10^100 bits (a closer estimation would be > 10^205, but that is irrelevant). Now, to express a number requires at > least one bit. What more is needed to see? WM is talking about accessibility of numbers, not their existence, which is quite a different issue.
From: Virgil on 14 Nov 2006 19:59 In article <1163506063.849027.169180(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. That effectively says that it is "necessary" in any geometry in which it is assumed, since a geometry is not Euclidean without it (or something equivalent to it, like the Playfair axiom). This is a lovely example of the circularity or WM's thought processes. > > 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. Thus WM isolates himself from mathematicians, for whom the discovery of non-Euclidean geometries was a giant step forward in mathematics.
From: Virgil on 14 Nov 2006 20:04 In article <1163506214.782141.176790(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > A finite initial segment of natural numbers includes > > its cardinal number. > > Rejected. Why does WM reject it? Does Wm claim that it is false? It is certainly true both for WM and for those who disagree with WM. The issue appears to be that WM will not be satisfied by what is generally acceptable as true, unless the word "finite" is deleted so as to make it false in ZF and NBG.
From: Dik T. Winter on 14 Nov 2006 20:43
In article <1163505343.116057.183460(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > 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 = 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. > > Not in your sense. He wrote, for instance to Killing, on April 5, > 1895: > Was Herr Veronese dar?ber in seiner Schrift giebt, halte ich f?r > Phantastereien und was er gegen mich darin vorbringt, ist unbegr?ndet. > > Ueber seine unendlich gro?en Zahlen sagt er, da? sie auf anderen > Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber > auf gar keinen Hypothesen sondern sind unmittelbar aus dem nat?rlichen > Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von > Willk?r, wie die endlichen ganzen Zahlen. > > You see: Gar keine Hypothesen. Cantor's axioms are not chosen but they > are necessary. Yes, he did regard it as such, that does not mean that he is right. In principle no axiom is necessary. But you need a few to have some start to work with. And when you add axioms to the basic set you will get more and more structure in what you have, untill you have added to many axioms. (Assuming of course that none of the axioms can be proven from the other axioms.) One of the basic researches in mathematics is what parts need which axioms, and what the result is when you drop axioms or replace them by other axioms. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |