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FROM MULTISETS TO SETS IN HOMOTOPY TYPE THEORY

  • HÅKON ROBBESTAD GYLTERUD (a1)

Abstract

We give a model of set theory based on multisets in homotopy type theory. The equality of the model is the identity type. The underlying type of iterative sets can be formulated in Martin-Löf type theory, without Higher Inductive Types (HITs), and is a sub-type of the underlying type of Aczel’s 1978 model of set theory in type theory. The Voevodsky Univalence Axiom and mere set quotients (a mild kind of HITs) are used to prove the axioms of constructive set theory for the model. We give an equivalence to the model provided in Chapter 10 of “Homotopy Type Theory” by the Univalent Foundations Program.

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[1]Aczel, P., The type theoretic interpretation of constructive set theory, Logic Colloquium ’77 (MacIntyre, A., Pacholski, L., and Paris, J., editors), North–Holland, Amsterdam, 1978, pp. 5566.
[2]Aczel, P. and Gambino, N., The generalised type-theoretic interpretation of constructive set theory, this JOURNAL, vol. 71 (2006), no. 1, pp. 67103.
[3]Gylterud, H. R., Multisets in Type Theory, preprint, 2016, arXiv:1610.08027.
[4]Martin-Löf, P., Intuitionistic Type Theory, Studies in Proof Theory, vol. 1, Bibliopolis, Naples, 1984.
[5]The Univalent Foundations Program, Homotopy Type Theory: Univalent Foundations of Mathematics, homotopytypetheory.org, Institute for Advanced Study, 2013.

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FROM MULTISETS TO SETS IN HOMOTOPY TYPE THEORY

  • HÅKON ROBBESTAD GYLTERUD (a1)

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