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Feb26-12, 11:13 AM
P: 206
It is definition (although well motivated). The usual notions of counting tend to break down when speaking of infinite sets. Two sets have the same cardinality iff there exists a 1-1 correspondence between the sets. So, for example, the set of all positive integers and the set of all even positive integers. One is a proper subset of the other yet they have the same cardinality. Map 1 to 2, 2 to 4, 3 to 6, etc. Asking a question about how many elements each set contains doesn't really make sense anymore in the way we traditionally think of counting a finite number of things.