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From: MoeBlee on 17 Oct 2006 13:30 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. All of this is covered in many a book on set theory and mentioned in many basic undergraduate texts in real analysis. MoeBlee
From: MoeBlee on 17 Oct 2006 13:33 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > You have an interpretation of a thought experiment that differs from > > the interpretation of other people. That doesn't make set theory > > inconsistent. It just makes set theory not suitable for your intuitions > > regarding the thought experiment. > > The inconsistency is that > 1) For the balls inserted until noon, you can find the result: It is > the set N. > 2) For the balls removed until noon, you can find the result: It is the > set N. > 3) For the balls remaining at noon, the same arguments of continuity > which lead to (1) and (2) cannot apply. > > This is the contradiction. No matter what the result (3) may be. In other words, just as I said, you have an interpretation of a thought experiment that differs from the interpretation of other people, but you don't mention a sentence in the language of set theory such that that sentence and its negation are theorems of set theory. MoeBlee
From: MoeBlee on 17 Oct 2006 13:35 mueckenh(a)rz.fh-augsburg.de wrote: > It is a rather silly accusation to speak of "intuition" in connection > with the observation that continuously accumulating numbers cannot lead > to an empty set. Well, if you'd offer a formal theory, then I'd evaluate it as opposed to your intutions regarding an infinitary thought experiment. MoeBlee
From: MoeBlee on 17 Oct 2006 13:41 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. > > All of this is covered in many a book on set theory and mentioned in > many basic undergraduate texts in real analysis. > > MoeBlee 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. MoeBlee
From: MoeBlee on 17 Oct 2006 13:46
MoeBlee wrote: > > 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, Correction: Instead of 'limit of the sum", that should be: the limit of the sequence of sums. MoeBlee |