A Complete Characterization of Choiceless Cardinality Comparison

We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets as a Boolean algebra together with the additional structure of comparing cardinality -- in the Cantorian sense of injections. What principles does one need to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization. This is joint work with Matthew Harrison-Trainor.