Here is Cantor's proof. Suppose that ℘S and S are the same size. Then, by the set-theoretic definition of “same size” (more correctly, ‘same cardinality’) there is a one-to-one surjection between S and ℘S. This means that there is a function that matches up each member of S with a unique member of ℘S so that there are no members of ℘S left over. Let us call this function, f. Then, if x is a member of S, f(x) is in ℘S. Now, since ℘S is the power set of S, it may be that x is in f(x) or it may not be in f(x).  Let us now define a set C: