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From: Arthur Fischer on 13 Apr 2005 12:36 worldsofsolution(a)yahoo.com wrote: > There is something I've been wondering about cantor's proof: the > decimal number generated to prove the contradiction, was taken to be a > real number. There is a tacit assumption that all decimal numbers > represent a real number. Does that not require a proof? Well, decimal expansions are just another way of representing infinite seies, in which case 0.d_1d_2d_3... is a "short-hand" notation for \sum_{i=1}^\infty d_i 10^{-i} (where, of course, the "infinite sum" is just the limit approached by the finite partial sums). Simple tests show that any such series is Cauchy, and by the completeness of the real line, the series converges (to a real number). The only "tacit assumption" made is the completeness of the real line, which most mathematicians would allow. __ Arthur
From: Torkel Franzen on 13 Apr 2005 12:47 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > Regrettably my time is limited. Fortunately, future time is unlimited! In the fullness of time, your decisive objections will carry the day, and all of this mathematical nonsense currently prevalent will be eradicated.
From: Randy Poe on 13 Apr 2005 12:47 Eckard Blumschein wrote: > On 4/13/2005 2:10 PM, David Kastrup wrote: > > Nonsense. Cantor never made such a list. > > At first he assumed that all reals are represented in his list. > Then he showed that at least one number is not contained in his list. Is proof by contradiction one of the things you don't understand? Do you know that if I begin a proof with: "Assume the square root of 2 is rational" that it does not mean I think the square root of 2 is rational? - Randy
From: David Kastrup on 13 Apr 2005 12:47 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 4/13/2005 2:10 PM, David Kastrup wrote: > >> You are babbling pseudophilosophical sophistic hogwash. That is simly >> irrelevant to the math. > > Please indicate if you will have anything to contribute. > Experts do not use this language. Kindergarten teachers do not use this language. Not all experts are qualified for that job. If you want an example for dealing with charlatans, you need not look further than your beloved Platon. Read the "Euthydemon". -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Will Twentyman on 13 Apr 2005 12:52
Eckard Blumschein wrote: > On 4/12/2005 10:57 PM, Will Twentyman wrote: > > >>Don't worry, I think several of us had already figured all that out. I >>think Eckard's problem is simply that he doesn't understand the concept >>of definition or proof. Intuition may inspire a line of reasoning, but >>is never a substitute for proof. There seem to be some insightful >>responses to his nonsense, though. > > I conclude that you did not read M280. You conclude incorrectly. I simply disagree with it at so many points that I consider it unlikely that we can come to any agreement on it. More specifically, reading it causes me to believe you do not understand what Cantor was doing or how mathematicians reason. I agree with you that Cantor's original notation may not have been tidily presented, but his fundamental concepts were sound and have been formalized. I note that you ignore ZF, ZFC, and other formalizations that may have eliminated any "rough edges" on Cantor's terminology or exposition in favor of the papers that serve as the basis for those works. Any particular reason why? I'll give one example where you are simply wrong: "Cýs infinite alephs only distinguish between countable and uncountable sets." The infinite alephs establish equivalence classes of sets which have a natural partial ordering based on surjections, and which also subdivide the the class of uncountable sets. -- Will Twentyman email: wtwentyman at copper dot net |