Well- and Non-Well-Founded Fregean Extensions

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege’s inconsistent Axiom V. We build on Boolos’s interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of the principle of limitation of size. After providing a complete structural description of all well-founded models, we turn to the non-wellfounded ones. We show how to build models in which foundation fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set, and also models of Aczel’s anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. There have been many recent and interesting attempts to develop much of standard set theory, by which we mean Zermelo-Fraenkel set theory plus the axioms of choice (ZFC), from some consistent modification of Frege’s original Axiom V in the framework of second-order logic. One prominent example occurs in [2], where George Boolos interprets a fragment of ZFC in the second-order theory whose only axiom is a variant of Axiom V based on von Neumann’s principle of limitation of size. More recently, Stewart Shapiro has surveyed in [5] efforts to extend the FregeBoolos strategy for developing standard set theory from variants of Axiom V that are acceptable from the point of view of neo-Fregeanism as championed by Crispin Wright and Bob Hale. Each of these attempts has been driven by a different overall aim, but what is of interest to us is how such developments illuminate the relation in which Cantorian sets, the objects of standard set theory, stand with respect to Fregean extensions as captured by consistent restrictions of Axiom V. The aim of this article is to further the comparison between Cantorian sets and Fregean extensions. Our point of departure are consistent modifications of Axiom V informed by the limitation of size doctrine. These principles determine a class of models in which most, and even all of the axioms of second-order Zermelo-Fraenkel set theory plus choice minus foundation (ZFC−) are satisfied. Some of these models are well-founded and may be of interest for those engaged in the neo-Fregean project to secure set theory within a theory of extensions. It is unclear whether a limitation of size modification of Axiom V may in fact serve as neo-Fregean foundation of set