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Communication Dans Un Congrès Année : 2006

Lambda-Z: Zermelo's Set Theory as a PTS with 4 Sorts

Alexandre Miquel
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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.
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Dates et versions

hal-00150886 , version 1 (03-06-2007)

Identifiants

  • HAL Id : hal-00150886 , version 1

Citer

Alexandre Miquel. Lambda-Z: Zermelo's Set Theory as a PTS with 4 Sorts. TYPES'04, 2006, Jouy-en-Josas, France. pp.232-251. ⟨hal-00150886⟩
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