Lambda-Z: Zermelo's Set Theory as a PTS with 4 Sorts
Résumé
We introduce a pure type system (PTS) lambda-Z 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 Lambda-Z via the `sets as pointed graphs' translation makes lambda-Z 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 lambda-Z to a (skolemised version of) Zermelo's set theory and by showing that both transformations commute via the axioms AFA and TC.
Domaines
Logique en informatique [cs.LO]
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