This re-instated what passes for faith in humanity

[Thecla]

You don't even need to neglect the repeats and it still works :)
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Well, one last tiny comment: if you don't ignore repeats you get a map of the natural numbers onto the rationals, but it's not one-to-one since different natural numbers map to the same rational, so that doesn't quite prove their cardinalities are equal, only that

card(rationals) <= card(natural numbers)

Of course, since the natural numbers are a subset of the rationals the reverse inequality holds, so they're actually equal.

But, for example, if you map the real numbers into the rationals by saying that x maps to x if x is rational and x maps to 0 if it's irrational, then you get a map of the reals onto the rationals, and in that case the cardinality of the reals is strictly greater than the cardinality of the rationals.