## Journal of Symbolic Logic

### The generalised type-theoretic interpretation of constructive set theory

#### Abstract

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive instead of being formulated via the propositions-as-types representation. 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.

#### Article information

Source
J. Symbolic Logic Volume 71, Issue 1 (2006), 67-103.

Dates
First available in Project Euclid: 22 February 2006

http://projecteuclid.org/euclid.jsl/1140641163

Digital Object Identifier
doi:10.2178/jsl/1140641163

Mathematical Reviews number (MathSciNet)
MR2210056

Zentralblatt MATH identifier
05038888

#### Citation

Gambino, Nicola; Aczel, Peter. The generalised type-theoretic interpretation of constructive set theory. J. Symbolic Logic 71 (2006), no. 1, 67--103. doi:10.2178/jsl/1140641163. http://projecteuclid.org/euclid.jsl/1140641163.

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