[jessaustin]

...only works for natural numbers...

That proof also works for the rationals with a suitable ordering. Example: 0, 1, -1, 2, -2, 1/2, -1/2, 3, -3, 1/3, -1/3, 2/3, etc....

[gnulinux]

Yes works for all enumerable set (i.e. all sets that have a bijection with natural numbers).