In mathematical set theory, a permutation model is a model of set theory with atoms constructed using a group of permutations of the atoms. Permutation models were introduced by Fraenkel (1922) and developed further by Mostowski (1938). One application is to show the independence of the axiom of choice from various other axioms in set theory with atoms.
Construction of permutation models
Suppose that A is a set of atoms, and G is a group of permutations of A. A normal filter of G is a collection F of subgroups of G such that
- G is in F
- The intersection of two elements of F is in F
- Any subgroup containing an element of F is in F
- Any conjugate of an element of F is in F
- The subgroup fixing any element of A is in F.
If V is a model of ZFA with A the set of atoms, then an element of V is called symmetric if the subgroup fixing it is in F, and is called hereditarily symmetric if it and all elements of its transitive closure are symmetric. The permutation model consists of all hereditarily symmetric elements, and is a model of ZFA.