Abstract

We adapt C. Freiling's axioms of symmetry 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles An, for n2, of increasing strength. Several equivalents of A2 are given. A2 is incompatible both with the foundation axiom and the antifoundation axioms AFA considered in Aczel . A hierarchy of symmetry degrees of preorderings is introduced and compared with An. Models are presented in which this hierarchy is strict. The main result of the paper is that a class X satisfies ¬An iff it has symmetry degree n−2