We prove a number of elementary facts about computability in partial combinatory algebras. We disprove a suggestion made by Kreisel about using Friedberg numberings to construct extensional pca’s. We then discuss separability and elements without total extensions. We relate this to Ershov’s notion of precompleteness, and we show that precomplete numberings are not 1–1 in general.

For every partial combinatory algebra, we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca’s, i.e., the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene’s first model is ${\omega _1^{\textit {CK}}}$ and that the closure ordinal of Kleene’s second model is $\omega _1$. We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of (...) pca’s. (shrink)

We consider so‐called extremal numberings that form the greatest or minimal degrees under the reducibility of all A‐computable numberings of a given family of subsets of, where A is an arbitrary oracle. Such numberings are very common in the literature and they are called universal and minimal A‐computable numberings, respectively. The main question of this paper is when a universal or a minimal A‐computable numbering satisfies the Recursion Theorem (with parameters). First we prove that the Turing degree of a set (...) A is hyperimmune if and only if every universal A‐computable numbering satisfies the Recursion Theorem. Next we prove that any universal A‐computable numbering satisfies the Recursion Theorem with parameters if A computes a non‐computable c.e. set. We also consider the property of precompleteness of universal numberings, which in turn is closely related to the Recursion Theorem. Ershov proved that a numbering is precomplete if and only if it satisfies the Recursion Theorem with parameters for partial computable functions. In this paper, we show that for a given A‐computable numbering, in the general case, the Recursion Theorem with parameters for total computable functions is not equivalent to the precompleteness of the numbering, even if it is universal. Finally we prove that if A is high, then any infinite A‐computable family has a minimal A‐computable numbering that satisfies the Recursion Theorem. (shrink)