The complexity of completions in partial combinatory algebra

We discuss the complexity of completions of partial combinatory algebras, in particular of Kleene's first model. Various completions of this model exist in the literature, but all of them have high complexity. We show that although there do not exist computable completions, there exists completions of low Turing degree. We use this construction to relate completions of Kleene's first model to complete extensions of PA. We also discuss the complexity of pcas defined from nonstandard models of PA.