  We exhibit a way of "forcing a functional to be an effective operation" for
arbitrary partial combinatory algebras (pcas). This gives a method of defining
new pcas from old ones for some fixed functional, where the new partial
functions can be viewed as computable relative to that functional. It is shown
that this generalizes a notion of computation relative to a functional as
defined by Kleene for the natural numbers. The construction can be used to
study subtoposes of the Effective Topos. We will do this for a particular
functional that forces every arithmetical set to be decidable. In this paper we
also prove the convenient result that the two definitions of a pca that are
common in the literature are essentially the same.

Logic