Abstract
Abstract 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 are no computable completions, there exist completions of low Turing degree. We use this construction to relate completions of Kleene’s first model to complete extensions of $\mathrm{PA}$ . We also discuss the complexity of pcas defined from nonstandard models of $\mathrm{PA}$ .
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