Which mathematical result shows some infinities are strictly larger than others?

Cantor's diagonal argument. Suppose you had a complete list of every real number between 0 and 1. Build a new number whose first digit differs from the first digit of the first entry, whose second differs from the second digit of the second, and so on. It differs from every number on the list in at least one place, so it is not on the list - and the list was assumed complete. There are therefore more reals than integers.

That is a strictly bigger infinity, and it is a proof of about four lines. Cantor went further and showed there is no largest infinity at all: the set of all subsets of any set is always bigger than the set itself, so the infinities go up forever.

The reaction was hostile. Kronecker, who had been his teacher, blocked publication and called him a corrupter of youth; Poincaré called the theory a disease. Cantor suffered severe depression from his forties and spent long periods in a sanatorium, and died in one in 1918. Hilbert's verdict is the one that stuck: no one shall expel us from the paradise that Cantor has created. His remaining question - whether an infinity sits between the integers and the reals - turns out to be undecidable from the standard axioms, proved in two halves by Gödel and Cohen.