Cantor Confusion
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From: Virgil on 2 Oct 2006 14:09 In article <3400f$4520e040$82a1e228$32660(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > My logic and my set theory get along famously. > > Please, Virgil, show us some of your famous work. > > Han de Bruijn Though I do not claim them as my "work", the logic I have adopted is mathematical logic and the set theory I have adopted is NBG, so you can easily look them up yourself.
From: Virgil on 2 Oct 2006 14:14 In article <e4ca$45210027$82a1e228$12053(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > I don't think there is any common sense about the infinite. > > Han de Bruijn Precisely why one should not try to judge such things by "common sense".
From: Virgil on 2 Oct 2006 14:18 In article <a7761$452120ff$82a1e228$22812(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > William Hughes wrote: > > > If both "there are an infinite number of balls in the vase at noon" > > and "there are no balls in the vase at noon" are equally > > good (or equally bad) common sense answers, > > you can't argue that the set theoretic answer is not > > the correct common sense answer. > > Common sense can only answer questions about the finite. It concludes > that the number of balls becomes larger and larger as we come closer > to noon. So extrapolating to noon itself can not result in zero balls. > > So far so good for common sense. Mathematics learns us that the limit > for time -> noon does not exist. And hence "noon" is a nonsense concept > in the context of this problem: the problem becomes ill-posed at noon. > > Han de Bruijn Mathematics may teach us all sorts of things, but it does not teach us that noon cannot exist, it merely teaches us that standard limit processes will not determine what the situation will be at noon.
From: mueckenh on 2 Oct 2006 14:21 William Hughes schrieb: > > Look, instead of the thought experiment constructed by Han and Tony you > > could also make the following thought experiment: Put 9 balls in the > > vase and put one ball in the urne. Logic says that the result cannot be > > different. > > > > No, this is a very different thought experiment. > Logic says the result will be different. > It matters not only how many balls are > added/removed, but also which balls. > > > If logic and set theory clash, abandon set theory. > > Indeed, but logic and set theory do not clash. > Set theory and intuition about infinite sets > clash. It is one of the worst reproaches at all if one can accuse a theory that its results depend on the symbols chosen to denote its elements. Regards, WM
From: mueckenh on 2 Oct 2006 14:22
Virgil schrieb: > In article <1159698057.723006.10500(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > That was later conflated to a proof about the reals. > > > > > > > > > > It was later shown that it could be modified to form a proof that the > > > > > set of all reals is uncountable. > > > > > > > > This was *not* "later shown", but at the very time of publishing in > > > > 1890/91 Cantor considered this very proof as the proof of the > > > > uncountability of he reals. > > > > > > > > Cantor, in the first paragraph: " Es läßt sich aber von jenem Satze > > > > [uncountability of the reals] ein viel einfacherer Beweis liefern, der > > > > unabhängig von der Betrachtung der Irrationalzahlen ist." > > > > My translation: "Here is a much simpler proof of the theorem > > > > [uncountability of the reals] which is independent of the reference to > > > > irrational numbers" > > > > > > As it is not clear that this sentence refers to any such theorem, I take > > > leave to doubt "Mueckenh"'s claim. > > > > Here is the full text: > > > > In dem Aufsatze, betitelt: Über eine Eigenschaft des Inbegriffs aller > > reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258), findet sich > > wohl zum ersten Male ein Beweis für den Satz, daß es unendliche > > Mannigfaltigkeiten gibt, die sich nicht gegenseitig eindeutig auf die > > Gesamtheit aller endlichen ganzen Zahlen 1, 2, 3, ..., nü, ... > > beziehen lassen, oder, wie ich mich auszudrücken pflege, die nicht die > > Mächtigkeit der Zahlenreihe 1, 2, 3, ..., nü, ... haben. Aus dem in > > § 2 Bewiesenen folgt nämlich ohne weiteres, daß beispielsweise die > > Gesamtheit aller reellen Zahlen eines beliebigen Intervalles > > (alpha...beta)sich nicht in der Reihenform > > > > w1, w2, ... wnü, ... > > > > darstellen läßt. > > Es läßt sich aber von jenem Satze ein viel einfacherer Beweis > > liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist. > > > > Regards, WM > > In the essay, calls: Over a characteristic of the epitome of all real > algebraic numbers (Journ. Math. Bd. 77, S. 258), a proof probably is for > the sentence for the first time that there is infinite variousnesses, > which cannot be referred mutually clearly to the whole of all finite > whole numbers of 1, 2, 3..., nue..., or, as I tend to be expressed, > those not the power of the zahlenreihe 1, 2, 3..., nue... to have. From > in 2 proving it follows easily that for example the whole of all real > numbers of any interval (alpha... beta) sich not in the row form w1, > w2... wnue... to represent leaves. However a much simpler proof can be > supplied by that sentence, which is independent of the view of the > irrational numbers.# A translation by a machine? But unimportant. Please look up what Cantor proved in this §2 of Journ. Math. Bd. 77, S. 258: There is nothing but numbers and rational functions of numbers. Regards, WM |