[seanhunter]

Yes. Trivially, uncountable infinities have to be larger than countable infinities do they not? The set of reals contains within it the set of integers for example but also contains a bunch of other stuff. (Not a mathematician obviously).

[schoen]

That argument turns out not to be enough! A counterexample is the set of rational numbers, which has the same cardinality as the natural numbers, even though the naturals are a proper subset of the rationals.

Some infinite sets can be put into one-to-one correspondence with some of their proper subsets!