Sets and Their Sizes

Abstract

This thesis presents an alternative to Cantor's theory of cardinality, insofar as that is understood as a theory of set size. The alternative is based on a general theory, ClassSize. ClassSize contains all sentences in the first order language with "subset" and "smaller-than" predicates which are true in all finite power sets. ClassSize is decidable but not finitely axiomatizable. Every infinite completion of ClassSize has a model over the power set of the natural numbers which satisfies an additional axiom, OUTPACING: If initial segments of A eventually become smaller than the corresponding initial segments of B, then A is smaller than B. Models which satisfy OUTPACING seem to accord with common intuitions about set size. In particular, they agree with the ordering suggested by the notion of "asymptotic density".