Manuel Lerman

Recursively Enumerable Sets Modulo Iterated Jumps and Extensions of Arslanov's Completeness Criterion

Let W e be the e th recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleenes recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that W e = W f ( e ) . In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets. Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper. Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete ( i.e. A has degree 0 ′) iff there is a function f recursive in A with no fixed point .

Some minimal pairs of α-recursively enumerable degrees

Exploitation of the model theoretic properties of G6dels constructible sets led in [61 to a generalization of the Friedberg-Muchnik finite injury (or priority) method from ¢o to every E l admissible ~. In order to generalize, it was necessary to sacrifice the standard indexing of arecursively enumerable sets, and hence of the requirements associated with finite injury arguments. For some as tile indexing was demonstrably not a-recursive. 131 gave an alternative view of [6] that centered on the nature of the indexing. This paper continues the study of indexing of requirements, and applies it to construct minimal pairs of a-recursively enumerable sets for some, but not all, ~x. The Friedberg-Muchnik solution of Posts problem generalizes in a trivial fashion to every 2; 2 admissible ordinal. All the complications of [31 and [61 resulted from forcing a 2;1 admissible ordinal a to do the work of a ~2 admissible ordinal. In this paper a is forced to do a much larger share of that work, and even

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta* be the Σ1-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α* if α*−M is undounded and for any α-r.e. setA, eitherA∩(α*-M) or (⇌*-A)∩(α*-M) is unbounded in α*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵL then it is shown that maximal sets do not exist.

The Universal Splitting Property. II

An r.e. set A of degree α is said to have the universal splitting property (U.S.P.) if for each r.e. degree β ≤ α, there is a splitting of A into disjoint r.e. sets B and C such that B has degree β. We show that any creative set has the U.S.P. and there are complete r.e. sets A which fail to have the U.S.P. We show that there are r.e. degrees a such that no r.e. set A of degree a has the U.S.P. We also explore the possible degrees of r.e. bases of r.e. vector spaces.

r-Maximal major subsets

AbstractThe question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂rmB) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ*(A) and ℒ*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂rmA as those with a Δ3 function that for each recursiveRi specifiesRi or

Degrees which do not bound minimal degrees

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Some Theorems on

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