Cantor Confusion
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From: Virgil on 19 Oct 2006 14:08 In article <d6483$45372f33$82a1e228$5556(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > imaginatorium(a)despammed.com wrote: > > > I expect all the non-cranks would agree with my statement, and find it > > astonishing. Cranks like yourself occasionally agree with all sorts of > > things, more or less by accident - so what? > > Cranks, cranks .. You're clearly running out of _arguments_, aren't you? > > Han de Bruijn As there are no arguments that will penetrate a crankhood as profound as HdB's, it would hardly matter if he did run out.
From: MoeBlee on 19 Oct 2006 14:08 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > MoeBlee wrote: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > You haven't yet noticed it? Each digit of the infinitely many digits of > > > > > > the diagonal number has the same weight or importance for the proof. In > > > > > > mathematics, the weight of the digits of reals is 10^(-n). Infinite > > > > > > sequences of digits with equal weight are undefined and devoid of > > > > > > meaning. > > > > > > > > > > The proof doesn't contradict the fact that the members of the sequence > > > > > are divided by greater and greater powers of ten. That fact is > > > > > mentioned in the previous proof showing the correspondence between the > > > > > sequences and real numbers. We prove that every sequence corresponds to > > > > > a real number where the real number is the limit of the sum of the > > > > > sequence made by taking greater and greater powers of ten in the > > > > > denominators, and that every real number corresponds to such a > > > > > sequence. THEN we proceed to the diagonal argument. > > > > > > > > > P.S. Again, if you disagree with the proof, then please just say what > > > > axiom or rule of inference you reject. In the meantime, again, there is > > > > no rational basis whatsoever for disputing that the argument does > > > > indicate a proof from the axioms per the rules of inference. > > > > > > Neglecting the powers 10^(-n) converts an infinite sequence which can > > > possibly yield a meaningful result into an impossible sequence, which > > > cannot be treated at all. But I believe your intuition will hinder you > > > accept that. > > > > You just went right past what I wrote. You just completely ignored what > > I wrote, which was in direct response to you. We do NOT neglect the > > exponentiation. > > You do. Otherwise there was no reason to explicitly exclude the case 1 > = 0.999. In real numbers we have this equation due to exponentiation. > With Cantor's diagonal we have not. The diagonal argument does not contradict that 1=.999... This is all discussed in such standard textbooks as Suppes's 'Axiomatic Set Theory' and other set theory textbooks and undergraduate introductions to real analysis. I'm willing to go through it step by step with you; but my fielding each of your ill-premised ad hoc red herrings is not a good way for you to learn the material. In other words, it makes much more sense to start with axioms and move FORWARD with them, covering every detail leading to the proof, rather then answering ill-premised red herring objections by explaining in REVERSE bits and pieces of the groundwork for the proof. MoeBlee
From: Virgil on 19 Oct 2006 14:12 In article <64afa$4537313e$82a1e228$6433(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > David Marcus wrote: > > > You said (repeatedly) that standard mathematics contains a > > contradiction, so please state the contradiction in standard > > mathematics. If you use new terms, please define them. > > How can you state a contradiction within catholicism while adopting the > standard dogmatism of that religion? As long as standard mathematics is > refusing to adopt the scientific method, it can not be "contradicted". And if mathematics were ever to be forced into that Procrustean bed, it would die.
From: Virgil on 19 Oct 2006 14:15 In article <986d0$45373235$82a1e228$6433(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > David Marcus wrote: > > > Dik T. Winter wrote: > > > >>I think that, compared to Cantor, in modern set theory potential and actual > >>infinity are split up again. The contents of the set N form only a > >>potential > >>infinity, on the other hand, the *size* is an actual infinity. > > > > I've never seen "potential infinity" or "actual infinity" in any > > textbook I've used. > > True. And you haven't seen any binary tree either. > > Han de Bruijn I have. There are all sorts of books and other references on binary trees. Google, for example, has over 1.5 million hits on "binary tree".
From: MoeBlee on 19 Oct 2006 14:16
mueckenh(a)rz.fh-augsburg.de wrote: > > I've never seen "potential infinity" or "actual infinity" in any > > textbook I've used. > > So you read not the right books or too few. 'potentially infinite' and 'actually infinite' are mentioned often in philosophy and history of mathematics. But would you please just refer to a single textbook of set theory, analysis, or calculus that gives mathematical definitions of 'potentially infinite' and 'actually infinite'? MoeBlee |