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 Re: hehe, Infinity Posted By: treellama Date: 12/15/06 4:12 p.m. In Response To: Re: hehe, Infinity (Forrest of B.org) : So by the time : you've finished counting the rational numbers that can be constructed out : of the complete set of natural numbers, you must have counted to a number : higher than any natural number. But, using Cantor's ordering, any rational number you can think of has a corresponding natural number. And vice versa. The only way there could be more rational numbers than natural numbers was if you could find a rational number without a corresponding natural number. Which you can't--the sets are the same size. : what exactly is meant by : "countable" if not in the hypothetical "...given that we : had infinite time to work with" sense, in which sense nothing but the : natural numbers themselves seem countable? Just that they can be arranged : into an ordered sequence, so that you would know how to even proceed in : the incompletable task of trying to count them? Yes. You certainly don't have to actually count them all. : But then I don't see how that's supposed to prove that all such countable sets have the same : cardinality. Because if you can order them and count them, thus labeling each one with a unique natural number, you show that there are just as many as there are natural numbers, and no more. Which is exactly what countable means.
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