Asaf Karagila (University of Leeds)

Measuring sizes without the Axiom of Choice, and how bad can it possibly get?

Picture yourself in a boat, on a river, with sets that can't be well-ordered and marmalade skies. How can you decide which ones are big and which ones are bigger? Sure, we can compare them by injections, or by surjections, but these are not total and some sets won't be comparable. One way to get around that is to use the Hartogs and Lindenbaum numbers which help us measure how large a set is, although in different ways, and to an extent, the relationship between them also tells us how far a set is from being well-ordered.