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FROM MULTISETS TO SETS IN HOMOTOPY TYPE THEORY

Published online by Cambridge University Press:  23 October 2018

HÅKON ROBBESTAD GYLTERUD
HÅKON ROBBESTAD GYLTERUD*
Affiliation:
DEPARTMENT OF INFORMATICS UNIVERSITY OF BERGEN POSTBOKS 7803 N-5020 BERGEN, NORWAYE-mail: hakon.gylterud@uib.no
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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.

Keywords

03B15constructive set theorytype theoryW-typeshomotopy type theoryhigher inductive typesmultisets

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 1132 - 1146
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

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The Univalent Foundations Program, Homotopy Type Theory: Univalent Foundations of Mathematics, homotopytypetheory.org, Institute for Advanced Study, 2013.Google Scholar