2026/07/26 by Junhong Chen, Daheng Ju
#math.LO #math.CO
We take the first step toward the study of set-theoretic principles below the axiom of choice AC and the axiom of foundation AF by studying Frucht's theorem, an ordinary mathematical theorem which is provable with either AC or 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.