Simple set

A recursively-enumerable set of natural numbers (cf. Enumerable set) whose complement is an immune set. Simple sets are intermediate in the sense of $m$-reducibility (cf. Recursive set theory) between solvable sets and creative sets. The latter are the largest among the enumerable sets in the sense of $m$-reducibility. Let $P$ be an arbitrary simple set, and let $K$ be an arbitrary creative set of natural numbers (e.g. the set of Gödel numbers of theorems of formal arithmetic); then there does not exist a general recursive function $f$ reducing $K$ to $P$, i.e. such that $$ x \in K \Leftrightarrow f(x) \in P \ . $$

Reducibility of $P$ to $K$ always takes place, but not one solvable set is reducible to $K$.

References

[1]	V.A. Uspenskii, "Leçons sur les fonctions calculables" , Hermann (1966) (Translated from Russian)
[2]	A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian)
[3]	H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165

Comments

A set is hypersimple if it is recursively enumerable and its complement is a hyperimmune set.

References

[a1]	Nies, André. Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford University Press (2009). ISBN 978-0-19-923076-1. Zbl 1169.03034