Notions of symmetry in set theory with classes Original Research Article

We adapt C. Freilingʹs axioms of symmetry (J. Symbolic Logic 51 (1986) 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles View the MathML source, for n⩾2, of increasing strength. Several equivalents of View the MathML source are given. View the MathML source is incompatible both with the foundation axiom and the antifoundation axioms View the MathML source considered in Aczel (Non Well Founded Sets, CSLI Lecture Notes, vol. 14, Stanford University, 1988). A hierarchy of symmetry degrees of preorderings (and of classes carrying such preorderings) is introduced and compared with View the MathML source. Models are presented in which this hierarchy is strict. The main result of the paper is that (modulo some choice principles) a class X satisfies View the MathML source iff it has symmetry degree n−2.