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 Re: hehe, Infinity Posted By: treellama Date: 12/15/06 12:18 p.m. In Response To: Re: hehe, Infinity (Forrest of B.org) : Hmm... I remember that the set of all integers and the set of all rationals : were supposedly the same size, though I was never thoroughly convinced by : the supposed proof of that. You can count rational numbers, the same as you can count natural numbers, just have to do it in the right order: 1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, 3/2, 4/1, 5/1, etc. : I'm pretty sure the set of reals is larger : than either of those, but it may not be the next-largest set; I don't : think there's consensus on what set is of cardinality aleph-one, only that : the set of all integers is of cardinality aleph-null. The set of reals is certainly bigger, because it contains the set of irrational numbers, which are uncountable. I don't think there's a way to prove that the set of reals has cardinality aleph one, but you can make a cardinality aleph one set by taking the set of all possible subsets of a cardinality aleph null set; so, the set of all subsets of natural numbers, would have cardinality 2^aleph null, which is equal to aleph one. I think, lol. I'm really not convinced why any of this is useful, aside from making the brain hurt.
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