Normalization of IZF with replacement

  • Authors:

  • Wojciech Moczydłowski

  • Affiliations:

  • Department of Computer Science, Cornell University, Ithaca, NY

  • Venue:
  • CSL'06 Proceedings of the 20th international conference on Computer Science Logic
  • Year:
  • 2006

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Abstract

IZF is a well investigated impredicative constructive version of Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF$_{R}^{\rm --}$ . We define a typed lambda calculus λZ corresponding to proofs in IZF$_{R}^{\rm --}$ according to the Curry-Howard isomorphism principle. Using realizability for IZF$_{R}^{\rm --}$ , we show weak normalization of λZ by employing a reduction-preserving erasure map from lambda terms to realizers. We use normalization to prove disjunction, numerical existence, set existence and term existence properties. An inner extensional model is used to show the properties for full, extensional IZFR.