Abstract
We exhibit a way of forcing a functional to be an effective opera- tion for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new par- tial 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.
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