Cantor Confusion
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From: mueckenh on 14 Nov 2006 06:37 Virgil schrieb: > While infinite collections in any physical sense are not possible, why > are imaginary infinities, such as sets of numbers must be, unimaginable? You cannot imagine the set (that would require infinitely many neurons), but you imagine a typical number with its properties and the envelope. (That's why you think a set is more than the collection of its elements.) > > How is an imagined set with no members any more actual that an imagined > set with every natural number as a member? To me they are both equally > figments of the imagination. You are right. A set with no members is a squared nonsense. It is convenient to assume its existence for formal reasons, but it does not exist, as Cantor admits. Cantor: It is useful to introduce a symbol which expresses the absence of points. O means, that the set has no point, i.e., strictly speaking it not present at all. (Es ist ferner zweckmäßig, ein Zeichen zu haben, welches die Abwesenheit von Punkten ausdrückt, wir wählen dazu den Buchstaben O; P == O bedeutet also, daß die Menge P keinen einzigen Punkt enthält, also streng genommen als solche gar nicht vorhanden ist.) > How is an imagined horse any more real that an imagined unicorn? > An imagined unicorn is real in your brain, at least those parts which you can imagine. You will not be able to imagine the contents of its stomach or all its hairs (Unicorns have hairs, because black holes have hairs and unicorns live in lack holes.) But I am sure, you are not able to imagine infinitely many unicorns. > I can more easily imagine an infinite set of finite naturals than one > which must contain any infinite natural. You cannot imagine all the naturals. The envelope is without value. > > I cannot even imagine what an infinite natural (as distinct from an > infinite ordinal) would be like. For such fantastic creations one should > appeal to such as TO. No infinite number exists and, of course, no infinite natural exists. Regards, WM
From: mueckenh on 14 Nov 2006 06:38 Virgil schrieb: > In article <1163430459.318473.317960(a)i42g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > If you add only one element to each column, you get the order type > > omega + 1 for the length of he matrix. > > The "length", being a cardinality rather than an ordinality, is > unaffected, since Card(omega) = Card(omega+1) The diagonal must have an order type. It must have the order type omega and the order type omega + 1 simultaneously, because it maps lines on columns. Regards, WM
From: mueckenh on 14 Nov 2006 06:41 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 darüber in seiner Schrift giebt, halte ich für Phantastereien und was er gegen mich darin vorbringt, ist unbegründet. Ueber seine unendlich großen Zahlen sagt er, daß sie auf anderen Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber auf gar keinen Hypothesen sondern sind unmittelbar aus dem natürlichen Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von Willkür, 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. Regards, WM
From: mueckenh on 14 Nov 2006 06:47 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? Regards, WM
From: mueckenh on 14 Nov 2006 06:55
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 =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 für die ganze Mannigfaltigkeitslehre aus. Daß es immer > > möglich ist, jede wohldefinierte Menge in die Form einer > > wohlgeordneten Menge zu bringen, auf dieses, wie mir scheint, > > grundlegende und folgenreiche, durch seine Allgemeingültigkeit > > besonders merkwürdige Denkgesetz werde ich in einer späteren > > Abhandlung zurückkommen." (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 darüber in seiner Schrift giebt, halte ich für Phantastereien und was er gegen mich darin vorbringt, ist unbegründet. Ueber seine unendlich großen Zahlen sagt er, daß sie auf anderen Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber auf gar keinen Hypothesen sondern sind unmittelbar aus dem natürlichen Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von Willkür, wie die endlichen ganzen Zahlen. You see: Gar keine Hypothesen. Cantor's axioms are not chosen but they are necessary. Regards, WM |