We work in the setting of Zermelo-Fraenkel set theory without assuming the
Axiom of Choice. We consider sets with the Boolean operations together with the
additional structure of comparing cardinality (in the Cantorian sense of
injections). What principles does one need to add to the laws of Boolean
algebra to reason not only about intersection, union, and complementation of
sets, but also about the relative size of sets? We give a complete
axiomatization.
A particularly interesting case is when one restricts to the Dedekind-finite
sets. In this case, one needs exactly the same principles as for reasoning
about imprecise probability comparisons, the central principle being
Generalized Finite Cancellation (which includes, as a special case,
division-by-m). In the general case, the central principle is a restricted
version of Generalized Finite Cancellation within Archimedean classes which we
call Covered Generalized Finite Cancellation.Comment: 25 page