λZ: Zermelo's set theory as a PTS with 4 sorts

Authors:
Alexandre Miquel
Affiliations:
PPS & Université Paris 7, Paris
Venue:
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
Year:
2004

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Abstract

We introduce a pure type system (PTS) λZ with four sorts and show that this PTS captures the proof-theoretic strength of Zermelo's set theory. For that, we show that the embedding of the language of set theory into λZ via the ‘sets as pointed graphs' translation makes λZ a conservative extension of IZ+AFA+TC (intuitionistic Zermelo's set theory plus Aczel's antifoundation axiom plus the axiom of transitive closure)—a theory which is equiconsistent to Zermelo's. The proof of conservativity is achieved by defining a retraction from λZ to a (skolemised version of) Zermelo's set theory and by showing that both transformations commute via the axioms AFA and TC.