Journal of Philosophical Logic 33 (5):437-465 (2004)

Gabriel Uzquiano
University of Southern California
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 a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded 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 as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership.
Keywords AFA  Axiom V  extensions  George Boolos  Gottlob Frege  New V  non-well-founded sets  second-order set theory  ZFC
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DOI 10.1023/B:LOGI.0000046129.22633.96
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References found in this work BETA

Iteration Again.George Boolos - 1989 - Philosophical Topics 17 (2):5-21.

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