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 Re: hehe, Infinity Posted By: treellama Date: 12/15/06 2:09 p.m. In Response To: Re: hehe, Infinity (Forrest of B.org) : Right, I got that, but that's the proof that I didn't find convincing. Why : does both being countable mean that they have the same cardinal value? It : seems like there should be exponentially more rational numbers than : integers... that the cardinality of set of rationals should be the square : of the cardinality of set of integers. Given there are n integers, there : should be n rational numbers with any given integer as it's numerator or : denominator; or in other words, there should be as many rationals on the : number line between any two integers, as there are integers. 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. : The : countability proof was shown to me graphically, as a grid of the natural : numbers over the natural numbers, and they counted diagonally from 1/1 on : down like you did. But it seems that, since a 3x3 grid has 3^2=9 boxes, : and a 4x4 grid had 4^2=16 boxes, and so on, an aleph-null-by-aleph-null : grid should have (aleph-null)^2 boxes. That holds true for any finite number n you can conceive, but we're talking about the cardinality of the entire infinite set. There's still exactly one natural number for each rational and vice versa, no matter how high you count, using that mapping. : For that matter, shouldn't the set of natural numbers be smaller than the set : set of integers? And shouldn't the set of prime numbers be smaller than : that? It's trivial to order and count the primes: 2, 3, 5, 7 1, 2, 3, 4 So, you can set it up so that for each prime number, there's one and only one natural number that corresponds. Similarly, for integers 0, 1, -1, 2, -2 1, 2, 3, 4, 5 So, all three sets have the same cardinality : I mean, they're all going to be "infinity" but we're : already talking different values of "infinity" so what's the : problem there? I'd think that aleph-null would be the cardinality of the : set of natural numbers, or maybe the primes, and everything else would be : based around that. Aleph null is the cardinality of the sets of all naturals, integers, rationals, and primes, since they all have the same cardinality. : I guess I just don't see why both being countable automatically makes two : sets have the same cardinality. Is there only one countable cardinality? I : gather the answer is supposedly "yes", but why? Is there only : one uncountable cardinality? I gather the answer is supposedly : "no", but why not? 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. : Maybe I'm just not understanding something about the concept of countability. Maybe not :)
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