Quote by adriank
It is true that R^{n} = R for all positive integers n. In fact, even R^{ω} (the set of all sequences of real numbers) has the same cardinality as R. (This can be shown using the HahnMazurkiewicz theorem.)

Thanks adriank. I was unable to find HahnMazurkiewicz on the net, but I found HahnBanach and the separation theorem of Mazur (which might be the same thing or similar). I guess the term "transfinite" is somewhat dated and such numbers are simply characterized as infinite.
It seems there are no infinite cardinals between aleph null and aleph one (the continuum hypothesis), but this idea depends on the Axiom of Choice. An alternative view which does not depend on the Axiom of Choice is based on the Dedekind infinite cardinal 'm' such that m+1=m, and alpha null
< m. However, it seems that for any infinite cardinal there must be a power set which is larger such as 2^(aleph null) that Hurkyl alluded to. Doesn't this number lie between aleph null and aleph one?