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Summary: THE GENERALISED TYPE-THEORETIC INTEPRETATION
OF CONSTRUCTIVE SET THEORY
NICOLA GAMBINO AND PETER ACZEL
Abstract. We present a generalisation of the type-theoretic interpre-
tation of constructive set theory into Martin-Lof type theory. The
original interpretation treated logic in Martin-Lof type theory via the
propositions-as-types interpretation. The generalisation involves replac-
ing Martin-Lof 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.
Contents
Introduction 1
1. Pure type theories 6
2. Logic-enriched type theories 9
3. The generalised type-theoretic interpretation 12
4. An analysis of the original type-theoretic interpretation 23
5. Reinterpreting logic 27
Appendix A. Type-theoretic rules 34
References 38
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