From type theory to setoids and back

Erik Palmgren

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arXiv · 1909.01414

From type theory to setoids and back

Erik Palmgren

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Abstract

A model of Martin-L\"of extensional type theory with universes is formalized in Agda, an interactive proof system based on Martin-L\"of intensional type theory. This may be understood, we claim, as a solution to the old problem of modelling the full extensional theory in the intensional theory. Types are interpreted as setoids, and the model is therefore a setoid model. We solve the problem of intepreting type universes by utilizing Aczel's type of iterative sets, and show how it can be made into a setoid of small setoids containing the necessary setoid constructions. In addition we interpret the bracket types of Awodey and Bauer. Further quotient types should be interpretable.

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BibTeX RIS

Erik Palmgren. 2019-09-03. From type theory to setoids and back. https://arxiv.org/abs/1909.01414

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