This thesis provides a computational interpretation of type theory validating Brouwer’s uniform-continuity principle that all functions from the Cantor space to natural numbers are uniformly continuous, so that type-theoretic proofs with the principle as an assumption have computational content. For this, we develop a variation of Johnstone’s topological topos, which consists of sheaves on a certain uniform-continuity site that is suitable for predicative, constructive reasoning. Our concrete sheaves can be described as sets equipped with a suitable continuity structure, which we call C-spaces, and their natural transformations can be regarded as continuous maps. The Kleene-Kreisel continuous functional can be calculated within the category of C-spaces. Our C-spaces form a locally cartesian closed category with a natural numbers object, and hence give models of Gödel’s system T and of dependent type theory. Moreover, the category has a fan functional that continuously compute moduli of uniform continuity, which validates the uniform-continuity principle formulated as a skolemized formula in system T and as a type via the Curry-Howard interpretation in dependent type theory. We emphasize that the construction of C-spaces and the verification of the uniform-continuity principles have been formalized in intensional Martin-Löf type theory in Agda notation.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:649340 |
| Date | January 2015 |
| Creators | Xu, Chuangjie |
| Publisher | University of Birmingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.bham.ac.uk//id/eprint/5967/ |
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