The most basic construction of an r.e. nonrecursive set—e.g. of the halting problem—proceeds by taking the diagonal of a recursive enumeration of all r.e. sets. We will answer the question of which r.e. sets can be constructed in this manner.If ψ is a computable numbering of some class of partial recursive functions, we define the diagonal of ψ to be the set Kψ ≔ {i ∈ ω ∣ ψi(i)↓}- It is well known that Kφ is creative if φ is a Gödelnumbering, and that for each creative set K there exists a Gödelnumbering φ such that K = Kφ. That is to say, the class of diagonals of Gödelnumberings is characterized as the class of creative sets. This class was shown to be elementary lattice theoretic (e.l.t.) by Harrington (see [So87, XV. 1.1]).We give a characterization of diagonals of arbitrary computable numberings of the class P1 of all partial recursive functions. To this end we introduce the notion of a semihyperhypersimple (shhs) set, which generalizes the notion of hyperhypersimplicity to nonsimple sets. It is shown that the diagonals of numberings of P1 are exactly the non-shhs sets. Then, properties of shhs sets are discussed. For example, for each nonrecursive r.e. set A there exists a nonrecursive shhs set B ≤TA, but not every r.e. T-degree contains a shhs set. These results build upon previous work by Downey and Stob [DSta].The question whether the property “shhs” is (elementary) lattice theoretic remains open. A positive answer would give both an analog of Harrington's result mentioned above, and a generalization of the well-known fact, due to Lachlan [La68], that hyperhypersimplicity is e.l.t. Therefore, we suspect that shhs sets turn out to be useful in the study of the lattice of r.e. sets.Previously, for several constructions from recursion theory the role of the underlying numbering of P1 was investigated; see Martin ([Ma66a] or [So87, V.4.1]) and Lachlan ([La75] or [Od89, III.9.2]) for Post's simple set, and Jockusch and Soare ([JS73]; cf. also [So87, XII.3.6, 3.7]) for Post's hypersimple set. However, only Gödelnumberings were considered. An explanation for the greater variety which arises when arbitrary numberings of P1 are admitted is provided by the fact that the index set of Gödelnumberings is less complex than the index set of all numberings of P1. The former is Σ1-complete; the latter is Π4-complete.
Isomorphism types of index sets of partial recursive functions
