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The Generalised Type-Theoretic Interpretation of Constructive Set Theory

Nicola Gambino and Peter Aczel
The Journal of Symbolic Logic
Vol. 71, No. 1 (Mar., 2006), pp. 67-103
Stable URL: http://www.jstor.org/stable/27588436
Page Count: 37
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The Generalised Type-Theoretic Interpretation of Constructive Set Theory
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Abstract

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.

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