Affordable Access

Publisher Website

Notions of symmetry in set theory with classes

Authors
Journal
Annals of Pure and Applied Logic
0168-0072
Publisher
Elsevier
Publication Date
Volume
106
Identifiers
DOI: 10.1016/s0168-0072(00)00028-2
Keywords
  • Proper Class
  • Non-Foundation
  • Symmetry Axioms
  • M-Symmetric Total Preordering
Disciplines
  • Logic
  • Mathematics

Abstract

Abstract 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 satisfies ¬ A n iff it has symmetry degree n−2.

There are no comments yet on this publication. Be the first to share your thoughts.