An uncountable countable set
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From: Virgil on 15 Jul 2006 15:20 In article <1152980141.301411.57580(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > 0.1 = 1/b, where b is the base > > > 0.11 = 1/b + 1/b^2, where b is the base > > > 0.111 = 1/b + 1/b^2 + 1/b^3 ,where b is the base > > > ... > > > 0.111... = 1/(b-1), where b is the base In order to justify the "0." part of "0.1" 0r "0.11" , etc., one must have a natural number base greater than 1, otherwise the "0." is redundant. > > > > > > The result is the same. Hence, if 0.111... exists, > > > it is in he list and is not in the list. It exists but not as any of its predecessors. > > > You may add by *+ the list below with the diagonal number exchanged by > (0 --> 1). You will see that the diagonal number, whether written down > explicitly as in the list above or to be introduced in the list below > there is no change of the sum. Hence the diagonal is already in the > list. Not in anyone else's interpretation of your "list".
From: David Hartley on 15 Jul 2006 16:24 In message <1152979000.961294.290560(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de writes .... >He says it: those and only those transformations which are realized by >a finite or infinite set of transpositions, i.e., interchanges of two >elements. >> >> > > > I know your interpretation is wrong. >> > > >> > > What part of my interpretation is wrong? >> > >> > The implication that Cantor's statement could be correct (if special >> > constraints were obeyed by permutations) with the implication that >> > infinity could exist. >> >> Can you provide a proof of that statement? Because you state just above: >> "Cantor's conclusion is correct, with no doubt, if infinite sets like >> the set of my transpositions do exist." Which contradicts what you state >> here. > >No. Cantor obviously was correct, if infinite sets would exist. But >they do not. So he was wrong. And you are wrong, as I have shown above >by Cantor's text which you misinterpreted grossly. Cantor's proposition (as you interpret it) leads directly to a contradiction in standard set theory. So it is disproved. To show that set theory is inconsistent you must also show that it can be proved. You cannot do this by arguing over quotations from Cantor. He does not appear to have offered a proof and he was not infallible. You claim it is obvious, so prove it yourself, with no more quotations, (except references to standard theorems}. -- David Hartley
From: mike4ty4 on 15 Jul 2006 16:53 mueckenh(a)rz.fh-augsburg.de wrote: > An uncountable countable set > > There is no bijective mapping f : |N --> M, > where M contains the set of all finite subsets of |N > and, in addition, the set K = {k e |N : k /e f(k)} of all natural > numbers k which are mapped on subsets not containing k. > > This shows M to be uncountable. > > Regards, WM What is "|N"? I've never seen that before. What does the pipe ( "|") on the left side represent?
From: Virgil on 15 Jul 2006 17:39 In article <1152996810.939256.87840(a)m73g2000cwd.googlegroups.com>, mike4ty4(a)yahoo.com wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > An uncountable countable set > > > > There is no bijective mapping f : |N --> M, > > where M contains the set of all finite subsets of |N > > and, in addition, the set K = {k e |N : k /e f(k)} of all natural > > numbers k which are mapped on subsets not containing k. > > > > This shows M to be uncountable. > > > > Regards, WM > > What is "|N"? I've never seen that before. What does the pipe > ( "|") on the left side represent? I think it is sometimes used to indicate that the following character should be printed in bold typeface. Thus those who customarily use bold typefaces to indicate certain mathematical objects in more flexible media do it whenever boldface is not available.
From: Dik T. Winter on 15 Jul 2006 19:41
In article <1152978585.858028.263870(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > In usual mathematics 0.999... is assumed to exist. > > > > It is not assumed. There is a definition floating around giving a meaning > > to that notation. > > Without the definition that notation makes no sense. > > And with that definition it makes no sense either, as the definition is > of he same kind as the squared circle. You do not believe in limits? Otherwise, why does the definition not make sense? Quote the definition for precisely that notation: 0.999... = lim{n -> oo} sum{k = 1..n} 9.10^(-k) what part of that definition makes no sense? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |