Designing a Calculational Proof of Cantor's Theorem

The one purpose of this little Note is to show that formal arguments need not be lengthy at all; on the contrary, they are often the most compact rendering of the argument. Its other purpose is to show the strong heuristic guidance that is available to us when we design such calculational proofs in sufficiently small, explicit steps. We illustrate our approach on Georg Cantor’s classic diagonalization argument [chosen because, at the time, it created a sensation]. Cantor’s purpose was to show that any set S is strictly smaller than its powerset ℘S (i.e., the set of all subsets of S). Because of the 1-1 correspondence between the elements of S and its singleton subsets, which are elements of ℘S, S is not larger than ℘S, and our proof can now be focussed on the “strictly”, i.e., we have to show that there is no 1-1 correspondence between S and ℘S. We can confine ourselves to non-empty S.