“Several infinite sets with cardinality Aleph 0 have been described. We have also seen several infinite sets with cardinality 2^ Aleph 0. The obvious question is whether there is anything strictly larger than 2 ^ Aleph 0. The answer is yes. The powerset of a set is strictly larger than the set. From this we can see that the powerset of (0,1), denoted powerset((0,1)), will not be in correspondence with (0,1). That is, the set of subsets of the unit interval (0,1) will be larger than (0,1). This set will have cardinality 2 ^ 2 Aleph 0. It is difficult to wrap one's mind around such a set. Try to write down some of the elements.

Of course, there is no reason to stop there. We can go on using the powerset function to describe sets of even higher cardinality. None of the sets in different levels of infinity can be put into correspondence with each other.”