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