Notions of symmetry in set theory with classes

✍ Scribed by Athanassios Tzouvaras

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 171 KB
• Volume: 106
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(00)00028-2

✦ Synopsis

We adapt C. Freiling's 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 A n , for n¿2, of increasing strength. Several equivalents of A 2 are given. A 2 is incompatible both with the foundation axiom and the antifoundation axioms AFA ∼ 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 A n . Models are presented in which this hierarchy is strict. The main result of the paper is that (modulo some choice principles) a class X satisÿes @A n i it has symmetry degree n -2.