An equivalent of the axiom of choice in finite models of the powerset axiom.
Alexander Abian, Wael A. Amin · Notre Dame Journal of Formal Logic · 1990
It is shown that in a finite model for the set-theoretical Powerset axiom every set s has a Choice set iff every set s has a Meet set Πs.Moreover, the Choice set of s is unique and is equal to Πs, where Πs is a singleton and Πs E s.Let (F, E) be a finite model for the set-theoretical Powerset axiom, i.e., in (F, E) every set has a powerset.For instance, let us consider the finite model (M, E) whose domain consists of the three sets a,b,c and where the E-relation is defined by: