Frucht's theorem and other set-theoretic principles below the axiom of choice and the axiom of foundation

Abstract

We take the first step toward the study of set-theoretic principles below the axiom of choice $\mathsf{AC}$ and the axiom of foundation $\mathsf{AF}$ by studying Frucht's theorem, an ordinary mathematical theorem which is provable with either $\mathsf{AC}$ or $\mathsf{AF}$ but not provable without both, and its variants. Specifically, we propose a number of such principles, study the relations between these principles and the standard axioms, and prove provability and unprovability results using (infinite) graph-theoretic constructions and permutation models, which draw a preliminary map of this new area of set theory.