The generalised type-theoretic interpretation of constructive set theory

  title={The generalised type-theoretic interpretation of constructive set theory},
  author={Peter Aczel and Nicola Gambino},
  journal={J. Symb. Log.},
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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