The Inconsistency of a Brouwerian Continuity Principle with the Curry-Howard Interpretation

If all functions (N → N) → N are continuous then 0 = 1. This is a theorem of intensional (and hence of extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Σ type via the Curry-Howard interpretation. With an intuitionistic notion of anonymous existence, defined as the propositional truncation of Σ, it is consistent that all such functions are continuous. A model is Johnstone’s topological topos. On the other hand, any of these two intuitionistic conceptions of existence give the same, consistent, notion of uniform continuity for functions (N → 2) → N, again valid in the topological topos. It is open whether the consistency of (uniform) continuity extends to homotopy type theory. The theorems of type theory informally proved here are also formally proved in Agda, but the development presented here is self-contained and doesn’t show Agda code.