Nonstandard Functional Interpretations and Categorical Models

Recently, the second author, Briseid and Safarik introduced nonstandard Dialectica, a functional interpretation that is capable of eliminating instances of familiar principles of nonstandard arithmetic including overspill, underspill, and generalisations to higher types from proofs. We show that, under few metatheoretical assumptions, the properties of this interpretation are mirrored by first order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren. In doing so, we also draw some new connections between nonstandard principles, and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller-Nahm interpretion with two different kinds of quantifiers (with and without computational meaning), similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica from this by weakening the computational content of the existential quantifiers – a process we call herbrandisation. We also define a constructive sheaf model mirroring this new functional interpretation and show that the process of herbrandisation has a clear meaning in terms of these sheaf models.