Size and Function

Bruno Whittle (Yale University)

6 Nov, 2pm, LH 302.

The talk will be about a question that is generally taken to be settled: are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results (which I of course do not dispute) answer the question given an almost universally accepted principle relating size to the existence of functions. The principle is: for any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the talk, however, is to argue that the question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in a position to know this principle. I will do this by looking at what seem to be the strongest reasons for thinking that we are in a position to know the principle, and arguing against them.