Generalizing Hartogs' Trichotomy Theorem

TL;DR

This paper demonstrates that the Axiom of Choice can be derived from weaker hypotheses involving the comparability of finite families of cardinals, extending Hartogs' classical theorem.

Contribution

It generalizes Hartogs' Trichotomy Theorem by showing that comparability of any finite subset of cardinals suffices to imply the Axiom of Choice.

Findings

Comparability of any pair of cardinals implies AC.

Finite families of cardinals with at least one comparable pair imply AC.

Weaker hypotheses than Hartogs' original assumption are sufficient.

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.