Generalizing Hartogs' Trichotomy Theorem

Abstract

A celebrated argument of F. Hartogs (1915) deduces the Axiom of Choice from the hypothesis of comparability for any pair of cardinals. We show how each of a sequence of seemingly much weaker hypotheses suffices. Fixing a finite number $k>1$, the Axiom of Choice follows if merely any family of $k$ cardinals contains at least one comparable pair.