Cantor Confusion
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From: Dik T. Winter on 30 Sep 2006 20:34 In article <1159620019.413262.312370(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > Cantor, in the first paragraph: " Es l=E4=DFt sich aber von jenem Satze > [uncountability of the reals] ein viel einfacherer Beweis liefern, der > unabh=E4ngig 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" Again you *deliberately* omit the first part of that paragraph where he wrote (paraphrased): "Paper n contains a proof of the theorem that there are sets that have a larger cardinality than the natural numbers". The "jenem Satze" (that theorem) can only refer to the theorem quoted, not to the theorem actually proven in his earlier paper. Note also that Zorn, in his annotations, actually *did* show in what way it could be modified to a proof about the reals. If it is written: "In an earlier paper [1], I did show that there are numbers that are not rational, in this paper I will give another proof of that theorem." that does not mean that "this" paper gives an easier proof of the theorem actually proven in the earlier paper, but a different proof of the theorem just mentioned. In English and Dutch (and also German, I think), a "that" always refers to the closest possible referent. (In this particular case, the earlier paper could be a proof that sqrt(2) was not rational while the current paper merely proves that there must be irrational numbers, without being particularly useful in proving that sqrt(2) is irrational.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 1 Oct 2006 06:17 Peter Webb schrieb: > >> For every real number x, there exists a list, L_x such > >> that x is a member of L_x > > > > Where do they exist? It is impossible for most real numbers even to > > name them. You cannot name or construct more than aleph_0 real numbers. > > Nevertheless you insist, that also the other ones, which have no names > > and no other identification properties, should have complete lists? > > This is a complete red herring. There is no question that the Real generated > by Cantor's proof is computable (r. e,) if the original list is, - other numbers cannot be in a real list - > as he gives > an explicit construction. Of course. That's why the diagonal proof only proves the existence of numbers which belong to a countable set i.e. the set of constructible reals. This proof proves in essence that the countable set of constructible real numbers is uncountable. A fine result of set theory. > > But if you like, you can schematically consider all real numbers by the > > infinite binary tree which contains them all represented by a countable > > set of nodes and edges. I have shown by a rational relation that the > > set of branches (corresponding to real numbers) is not larger than the > > set of edges. > > > > Then the set of edges isn't countable either. The set of edges *is* countable. It is the simple method used for the square rational numbers, here adopted to a triangle: /1 \2 /3\4 /5\6 .... Obviously there remains no edge without a natural number. Regards, WM
From: mueckenh on 1 Oct 2006 06:20 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
From: Ross A. Finlayson on 1 Oct 2006 06:47 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 I have one fishing rod, it's a steelhead rod. My Dad made it. I broke, the tip, it costs, $2.50 to fix the steelhead tip in Kamiah, Kam, ee, IIII, Idaho, it only costs three dollars to fix your fishing pole. Reels are pre-made like your car in manufacturing units. Biggest fishes in die Worldt. Quel'che ch'mism 'expresmo! My, my relative, she has blue eyes. Me, I have to parents with brown eyes whose both parents had blue eyes. Did you know I work in a hospital? I'm a psychiatrist for doctors. Luckily, I have lots of spare time. Mueckenheim! Hallo. Du bist Deutscher. Aber, meing grandfatter, ettu tue grandvatter, en der lange deutsche, ... Thanks Wolfgang, Ross
From: Poker Joker on 1 Oct 2006 08:57
"Arturo Magidin" <magidin(a)math.berkeley.edu> wrote in message news:efmf1m$c88$1(a)agate.berkeley.edu... >>So if considering a single specific list >>shows a flaw, then looking at ANY (ALL of them) list doesn't >>help. > .. since no flaw has been exhibited by looking at any specific > list (and "specific" in this case must mean explicit and specific, not > a putative list with putative properties whose existence cannot be > established a priori; otherwise, we might just say "take a list for > which the argument does not work", which is of course nonsense), > discussions about this are a waste of time. First you obfuscate the discussion by saying that specific cases don't matter. Now you seem to imply they do, but in this discussion they still don't because of the obfustated argument that somehow they don't. After all, you never showed how step #2 isn't self-referential in the case that the process input is all real numbers. Now I understand how someone can reject a fields medal. |