Extremal numberings and fixed point theorems

Abstract

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 $N$, 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.