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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)

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.

Keywords

Transitive Closure Mutual Induction Conservative Extension Object Term Pointed Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

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

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