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Sets in types, types in sets

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Theoretical Aspects of Computer Software (TACS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1281))

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Abstract

We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Fraenkel set theory and the opposite way. More precisely, we actually construct two families of encodings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r.t. expressive power and thus are essentially equivalent. Both encodings are quite elementary: type theory is interpreted in set theory through a generalization of Coquand's simple proof-irrelevance interpretation. Set theory is encoded in type theory using a variant of Aczel's encoding; we have formally checked this last part using the Coq proof assistant.

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Authors and Affiliations

  1. INRIA-Rocquencourt, BP 105, 78 153, Le Chesnay Cedex, France

    Benjamin Werner

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  1. Benjamin Werner
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Martín Abadi Takayasu Ito

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© 1997 Springer-Verlag Berlin Heidelberg

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Werner, B. (1997). Sets in types, types in sets. In: Abadi, M., Ito, T. (eds) Theoretical Aspects of Computer Software. TACS 1997. Lecture Notes in Computer Science, vol 1281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014566

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  • DOI: https://doi.org/10.1007/BFb0014566

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63388-4

  • Online ISBN: 978-3-540-69530-1

  • eBook Packages: Springer Book Archive

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