Relaxing the Fraenkel-Mostowski Set Theory
Andrei Alexandru and Gabriel Ciobanu
We study the PA-sets which are (Zermelo-Fraenkel) sets equipped with permutation actions of the group of all bijections of a fixed set A of atoms. These PA-sets are related to SA-sets (sets with permutation actions), and also to nominal sets. We prove that some particular PA-sets have similar properties with nominal sets, although we ignore the finite support requirement. Then we present the Relaxed Fraenkel-Mostowski set theory by weakening the finite support axiom in Fraenkel-Mostowski set theory. In this relaxed version it is not required the existence of a finite support for all the higher-order constructions as in Fraenkel-Mostowski theory, but only for the elements of the powerset of an amorphous set of atoms. Finite-Cofinite Mathematics is presented as a refinement of Zermelo-Fraenkel with atoms formed only by either finite or cofinite atomic sets. Finally, we present a comparison between Fraenkel-Mostowski set theory and other related theories (including Relaxed Fraenkel-Mostowski theory and Finite-Cofinite Mathematics).
Keywords: Fraenkel-Mostowski set theory, relaxed finite support axiom, amorphous sets, finite-cofinite mathematics