Cantor Confusion
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From: Tony Orlow on 8 Jan 2007 10:02 Dave Seaman wrote: > On Mon, 08 Jan 2007 04:48:49 EST, Andy Smith wrote: >> A slightly different question. > >> A real point, such as pi or e has a genuine existance >> independent of its e.g. binary representation. Also integers >> e.g. 0 = 0.00... does not need to be defined as the limit point >> between 1/2^n and -1/2^n. > > You might look at > <http://www.cs.uwaterloo.ca/~alopez-o/math-faq/node11.html#SECTION00320000000000000000>, > concerning the construction of the number systems. > >> What about .11111.... ? That has a definite existance on >> th line - can one define that as the point that has no >> zeroes in its binary representation? > > Oh? What if I write it as 0.111...? > >> If so then one can argue that .1111... is different from >> 1.0000... and that there are no rationals between these >> two pints (because any rational must have a terminating >> string of 0000... or a repeating string which is other >> than all ones? > > A number is not a digit string. The fact that there are no rationals > between two reals constitutes proof that those reals are identical. > See the reference above for an explanation of Dedekind cuts. > > What if one of those reals is a rational itself? Is there necessarily a rational between every rational and every real, or are there irrational reals "adjacent" to rational reals?
From: mueckenh on 8 Jan 2007 10:50 William Hughes schrieb: > > That L_D does not exist. > > Assume a line that contains any element that can > be shown to be in the diagonal exists. Otherwise the diagonal could not exist, because it is, by definition, the union of line ends in the EIT. > Call > this line L_D. > > A: L_D is a line, therefore L_D has a largest > element. > > B: L_D contains any element that can be shown to exist > in the diagonal, therefore L_D does not have a largest > element. > > Contradiction. Therefore L_D does not exist. Therefore the union of line ends including the end of L_D does not exist. Fine. There are no infinite sets. There is only potential infinity. Every line including the diagonal has a greatest element until the existence of a greater one is shown. Regards, WM
From: Franziska Neugebauer on 8 Jan 2007 12:07 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: >> >> That L_D does not exist. >> >> Assume a line that contains any element that can >> be shown to be in the diagonal exists. [(p)] Wolfgang, are you familiar with reductio ad absurdum? In short: p & (q & ~q) -> ~p > Otherwise the diagonal could not exist, because it is, by definition, > the union of line ends in the EIT. Straw man. After _assuming_ (p) there is no need to justify the assumption. >> Call this line L_D. >> >> A: L_D is a line, therefore L_D has a largest >> element. [(q)] >> >> B: L_D contains any element that can be shown to exist >> in the diagonal, therefore L_D does not have a largest >> element. [(~q)] >> >> Contradiction. Therefore L_D does not exist. That is correct. > Therefore the union of line ends including the end of L_D does not > exist. The result of the William's proof is ~p: There is no such L_D. F. N. -- xyz
From: William Hughes on 8 Jan 2007 12:30 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > That L_D does not exist. > > > > Assume a line that contains any element that can > > be shown to be in the diagonal exists. > > Otherwise the diagonal could not exist, because it is, by definition, > the union of line ends in the EIT. > No. The diagonal is not a line. The union of line ends is not a line. The union of line ends (diagonal) is a potentially infinite set. A line is not a potentially infinite set. > > Call > > this line L_D. > > > > A: L_D is a line, therefore L_D has a largest > > element. > > > > B: L_D contains any element that can be shown to exist > > in the diagonal, therefore L_D does not have a largest > > element. > > > > Contradiction. Therefore L_D does not exist. > > Therefore the union of line ends including the end of L_D does not > exist. The union of line ends is not a line. L_D does not exist so the union of line ends including the end of L_D does not exist. > Fine. There are no infinite sets. There is only potential > infinity. Every line including the diagonal The diagonal is not a line. The diagonal is a potentially infinite set. A line is not a potentially infinite set > has a greatest element > until the existence of a greater one is shown. However, and contrary to your repeated claim, there is no single line which contains every element that can be shown to be in the diagonal. - William Hughes
From: David Marcus on 8 Jan 2007 13:34
Andy Smith wrote: > I do now see what you mean, but it still strikes me as a > bit Humpty Dumpty - " aword means what I see it means, > nothing more and nothing less". Of course it is Humpty Dumpty! Lewis Carroll was a mathematician! -- David Marcus |