λZ: Zermelo’s Set Theory as a PTS with 4 Sorts

  • Alexandre Miquel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3839)


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.


Transitive Closure Mutual Induction Conservative Extension Object Term Pointed Graph 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexandre Miquel
    • 1
  1. 1.PPS & Université Paris 7Paris

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