Realizability with a local operator of A.M. Pitts

We study a notion of realizability with a local operator JJ which was first considered by A.M. Pitts in his thesis [7]. Using the Suslin–Kleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical (Δ11) functions.We show that there is a realizability interpretation of nonstandard arithmetic, which, despite its classical character, lives in a very non-classical universe, where the Uniformity Principle holds and König's Lemma fails. We conjecture that the local operator gives a useful indexing of the hyperarithmetical functions.