A Constructive Model of Uniform Continuity

We construct a continuous model of Godel’s system T and its logic HA ω in which all functions from the Cantor space 2ℕ to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in particular, we can compute moduli of uniform continuity of T-definable functions 2ℕ → ℕ. Moreover, the model has a continuous Fan functional of type (2ℕ → ℕ) → ℕ that calculates moduli of uniform continuity. We work with sheaves, and with a full subcategory of concrete sheaves that can be presented as sets with structure, which can be regarded as spaces, and whose natural transformations can be regarded as continuous maps.

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