In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all (...) of these models, the algebraic closure of a set is nontrivial., is given in §1, however in vector spaces, cl is just the subspace generated byS, in Boolean algebras, cl is just the subalgebra generated byS, and in algebraically closed fields, cl is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings. (shrink)

This paper contributes to the question of under which conditions recursively enumerable sets with isomorphic lattices of recursively enumerable supersets are automorphic in the lattice of all recursively enumerable sets. We show that hyperhypersimple sets (i.e. sets where the recursively enumerable supersets form a Boolean algebra) are automorphic if there is a Σ 0 3 -definable isomorphism between their lattices of supersets. Lerman, Shore and Soare have shown that this is not true if one replaces Σ 0 3 by Σ (...) 0 4. (shrink)

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose many-one degrees are pairwise (...) incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′. The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈, where M is a maximal set, and M is its complement. We showed that / 1 ≡ / 2 if, and only if, 1 = m 2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree. (shrink)

Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X', its Turning jump, is recursive in $\varnothing'$ and high if X' computes $\varnothing''$ . Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of $A \bigoplus W$ is recursive in the jump of A. We prove that there are no (...) deep degrees other than the recursive one. Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turning functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that $(W \bigoplus A)'$ is forced to disagree with Φ(-; A'). The conversion has some ambiguity; in particular, A cannot be found uniformly from W. We also show that there is a "moderately" deep degree: There is a low nonzero degree whose join with any other low degree is not high. (shrink)

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show (...) that for all n ≥ 2, there exists a Δ 0 n+1 n-cohesive set. We improve this result for n = 2 by showing that there is a Π 0 2 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0 (n+1) and also characterize the jumps of the n-r-cohesive degrees. (shrink)

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, $\varepsilon = (\{W_e\}_{e\in \omega}, \subseteq)$ . We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has (...) a certain "slowness" property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ∈ ε there exists B in the orbit of A such that X ≤ T B under relative Turing computability (≤ T ). We produce B using the Δ 0 3 -automorphism method we introduced earlier. (shrink)

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, (...) ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable. We use A* to denote the equivalent class of A under the ideal of finite sets. (shrink)

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, (...) ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable. We use A* to denote the equivalent class of A under the ideal of finite sets. (shrink)

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class $superhigh^\diamond$ and show that it contains some, but not all, of the noncomputable K-trivial sets.

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us to better understand the${\cal D}$-maximal (...) sets, we use them to show that several classes of${\cal D}$-maximal sets break into infinitely many orbits. (shrink)

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b' = 0'). This extends Csima's result that every CAD theory has a low prime model. We (...) also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low₂ (c" > 0"), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b' is any degree c.e. in c with d' ≤ b'. As a corollary, we show that for any degree c with 0 < c < 0', every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models. (shrink)

A Turing degree a is said to be almost everywhere dominating if, for almost all $X \in 2^{\omega}$ with respect to the "fair coin" probability measure on $2^{\omega}$ , and for all g: $\omega \rightarrow \omega$ Turing reducible to X, there exists f: $\omega \rightarrow \omega$ of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in (...) the reverse mathematics of measure theory. (shrink)

Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us to better understand the${\cal D}$-maximal (...) sets, we use them to show that several classes of${\cal D}$-maximal sets break into infinitely many orbits. (shrink)

Given a reducibility ⩽r, we say that an infinite set A is r-introimmune if A is not r-reducible to any of its subsets B with |A\B| = ∞. We consider the many-one reducibility ⩽m and we prove the existence of a low1 m-introimmune set in Π01 and the existence of a low1 bi-m-introimmune set.

Let ℰΠ denote the collection of all Π01 classes, ordered by inclusion. A collection of Turing degrees.

A Schnorr test relative to some oracle A may informally be called “universal” if it covers all Schnorr tests. Since no true universal Schnorr test exists, such an A cannot be computable. We prove that the sets with this property are exactly those with high Turing degree. Our method is closely related to the proof of Terwijn and Zambella’s characterization of the oracles which are low for Schnorr tests. We also consider the oracles which compute relativized Schnorr tests with the (...) weaker property of covering all computable reals. The degrees of these oracles strictly include the hyperimmune degrees and are strictly included in the degrees not computably traceable. (shrink)

We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...) _\alpha ^0 $ orbit (from the proof). (shrink)

There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and its dual, the prediction number, as well as two versions of the rearrangement number. When translated appropriately, these yield four new highness (...) notions. We will define these new notions, show some of their basic properties and place them in the computability-theoretic version of Cichoń's diagram. (shrink)

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.

We investigate the orbit of a low computably enumerable set under automorphisms of the partial order of c.e. sets under inclusion. Given an arbitrary low c.e. set A and an arbitrary noncomputable c.e. set C, we use the New Extension Theorem of Soare to construct an automorphism of mapping A to a set B such that CTB. Thus, the orbit in of the low set A cannot be contained in the upper cone above C. This complements a result of Harrington, (...) who showed that the orbit of a noncomputable c.e. set cannot be contained in the lower cone below any incomplete c.e. set. (shrink)

In this paper we show that there is a pair of superhigh r.e. degree that forms a minimal pair. An analysis of the proof shows that a critical ingredient is the growth rates of certain order functions. This leads us to investigate certain high r.e. degrees, which resemble ∅′ very closely in terms of ∅′-jump traceability. In particular, we will construct an ultrahigh degree which is cappable.