Ooh, I know this one.
The relationship between aleph-1 and c is called is the Continuum Hypothesis. It's now known to be independent: you can choose it to be true or false, and either is consistent with standard mathematics.
The next infinity after aleph-0 is called aleph-1, then come aleph-2, aleph-3, ..., aleph-aleph-0, ...
Nothing is known that has size aleph-1 (or any of those others bigger than aleph-0).
The set of integers has the size aleph-0. The set of real numbers has a size called c (meaning continuum), bigger than aleph-0. The set of all subsets of the integers has a size that can be expressed as 2^aleph-0 (two to the power of aleph nought), which is also bigger than aleph-0.
So we have three infinities bigger than the smallest one: aleph-1, c, and 2^aleph-0. You can prove c = 2^aleph-0. Since aleph-1 is the next biggest, either c = aleph-1 or c > aleph-1. The Continuum Hypothesis, posed by Cantor, is that they're equal.
In 1938 Gödel proved it would be consistent if the CH was true. In 1963 Paul Cohen proved it would be consistent if the contrary was true (i.e. there were infinities strictly between aleph-0 and c).
P.S. Who needs infinities? They're mathematicians: when they get to nought bottles they keep going to -1. :-)
