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Multisets in type theory


A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin–Löf Type Theory, in the presence of Voevodsky’s Univalence Axiom.

In his 1978 paper, “the type theoretic interpretation of constructive set theory” Aczel introduced a model of constructive set theory in type theory, using a W-type quantifying over a universe, and an inductively defined equivalence relation on it. Our investigation takes this W-type and instead considers the identity type on it, which can be computed from the univalence axiom. Our thesis is that this gives a model of multisets. In order to demonstrate this, we adapt axioms of constructive set theory to multisets, and show that they hold for our model.

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[1]Aczel, P.. The type theoretic interpretation of constructive set theory. Logic Colloquirm 77, Ed. by MacIntyre, A., Pacholski, L. and Paris, J.. (North–Holland, Amsterdam-New York, 1978) pp 5566
[2]Aczel, P. and Michael, R.. Notes on Constructive Set Theory. Tech. Rep. Institut Mittag–Leffler (2001).
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[5]Danielsson, N. A.. Positive h-levels are closed under W., (2012).
[6]Escardó, M. H.. A self-contained, brief and complete formulation of Voevodsky’s univalence axiom. (2018) arXiv:1803.02294.
[7]Gylterud, H. R.. Formalisation of iterative multisets and sets in Agda, (2016) URL:
[8]Martin–Löf, P.. Intuitionistic type theory. Notes by Giovanni Sambin. Vol. 1. Studies in Proof Theory (Bibliopolis, Naples, (1984)) pp. iv+91.
[9]Nordström, B., Petersson, K. and Smith, J. M.. Programming in Martin–Löf’s Type Theory. (1990).
[10]Rado, R.. The cardinal module and some theorems on families of sets. Annali di Matematica Pura ed Applicata 102.1 (1975), pp. 135154.
[11]The Univalent Foundations Program (2013). Homotopy type theory: univalent foundations of mathematics. Institute for Advanced Study (2013).
[12]Wiener, N.. A simplification of the logic of relations. Proc. of Camb. Phil. Soc. 17, (1917), pp. 387390.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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