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 Re: hehe, Infinity Posted By: Forrest of B.org Date: 12/15/06 3:15 p.m. In Response To: Re: hehe, Infinity (treellama) : You would think that, but we're mapping one natural number to one rational : number here; there's a bijection. For each rational number in that list, : there's exactly one natural number that goes with it, and the reverse is : true. And as many as you can ever count of one, there's that many you can : count of the other. So, they're the same cardinality. : Because if you can order and count the members of an infinite set, you can : show there's a bijection between natural numbers, and the members of that : set. I guess my problem with accepting this is that it seems like, for any amount of rational numbers, by the time you've counted them all, you've counted to a natural number higher than any thus far used in any of those rationals. E.g. if you count all rationals constructed of only 1 and 2: 1: 1/1 2: 1/2 3: 2/1 4: 2/2 You now have two new natural numbers, 3 and 4, that could be used to construct a bunch more rationals (namely 1/3, 2/3, 3/3, 4/3, 3/1, 3/2, 3/4, 1/4, 2/4, 4/4, 4/1, and 4/2, in no particular order). So then by the time you've counted all them, you've got natural numbers up to 16, so that's 12 more natural numbers to construct rational numbers out of, after which point you'll have counted up to 256, and so on. 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. Which, now that I say it, sounds like a contradiction, which should mean that you can't ever count that high. Which of course you practically can't since it would take forever to do so no matter how fast you could count, but then that makes me ponder, 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? But then I don't see how that's supposed to prove that all such countable sets have the same cardinality. Mind you, I'm not asking you, Mr. Music Major, to answer all these things. Just kind of mumbling my confusions out loud. Though if you (or any one else) have answers, please, let me know. Postscript: a picture just popped into my head of two curves, both of which hit an asymptote at the same x-value, but one of which increases much more quickly; so that at x=1, they have only slightly different y values, and x=2, they have bigger differences in y values, and x=3 they have vastly different y values, and so on... but both reach infinity at, say, x=7. Is that even possible? Is that something like what's happening here; even though the cardinality of a subset of rationals is exponentially greater than the cardinality of the subset of naturals from which they're constructed, somehow both sets reach infinity "at the same time"?
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