Iain: There is a branch of math named “Set Theory”; and in it the following odd fact does indeed arise: ***IF*** you admit the concept of an infinite set (a set whose cardinal number is not diminished if you remove N members of the set, for any natural number N), ***THEN*** it is possible to construct two such sets whose members cannot be matched off one-to-one with each other. One of these sets would be “bigger” than the other, if you like, by analogy with the fact that a set of seven objects can’t be matched off one-to-one with a set of three objects because it is bigger.

Analogy is all it is, though, and the philosophical waters here are turbid. This whole zone is still controversial even after 120 years. You can find mathematicians of the “constructivist” or “intuitionist” schools–I actually know one–who will NOT admit the concept, or will admit it only under heavy and disabling qualifications.

Here is the guy who started this line of inquiry. Note where he ended up.