Alistair H. Lachlan

On strongly minimal sets

The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaughts conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.

The d.r.e. degrees are not dense

Abstract By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n -r.e. degrees ( n ⩾2) and of the ω-r.e. degrees.

Bounding Minimal Pairs

A minimal pair of recursively enumerable (r.e.) degrees is a pair of degrees a, b of nonrecursive r.e. sets with the property that if c ≤ a and c ≤ b then c = 0. Lachlan [2] and Yates [4] independently proved the existence of minimal pairs. It was natural to ask whether for an arbitrary nonzero r.e. degree c there is a minimal pair a, b with a ≤ c and b ≤ c . In 1971 Lachlan and Ladner proved that a minimal pair below c cannot be obtained in a uniformly effective way from c for r.e. c ≠ 0. but the result was never published. More recently Cooper [1] showed that if c is r.e. and c ′ = 0″ then there is a minimal pair below c . In this paper we prove two results: Theorem 1. There exists a nonzero r.e. degree with no minimal pair below it . Theorem 2. There exists a nonzero r.e. degree c such that, if d is r.e. and 0 . The second theorem is a straightforward variation on the original minimal pair construction, but the proof of the first theorem has some novel features. After some preliminaries in §1, the first theorem is proved in §2 and the second in §3. I am grateful to Richard Ladner who collaborated with me during the first phase of work on this paper as witnessed by our joint abstract [3]. The many discussions we had about the construction required in Theorem 1 were of great help to me.

Then-rea enumeration degrees are dense

The notion of enumeration (or simply e-) reducibility between sets as formalized by Rogers even in early versions of (1967) [see for example Friedberg and Rogers (1959)] captures the natural concept of computing a set A from another one B with only positive membership information about B being available. One can view the procedure as a computation determined by a finite set of instructions except that from time to time an outside source informs the computing agent that certain numbers are in B. We say that A is e-reducible to B if there is such a procedure which, when supplied with all the members of B (in any order and at any time during its computation) as inputs, correctly enumerates A, i.e. lists precisely the members of A as its outputs. By identifying partial functions with their graphs, this notion includes the basic reducibility on partial functions introduced by Kleene (1952) and studied in Myhill (1961). When further restricted to the (graphs) of total functions, it coincides with the standard notion of Turing reducibility. In its unrestricted form, it is connected with the general problem of modeling computation from partial information, self application and models of the k-calculus. We refer the reader to Cooper (1990) for a general survey of enumeration reducibility and an extensive bibliography of related articles. To fix our notation, we present the formal definitions of enumeration operators, reducibilities and degrees as in Rogers (1967, Sect. 9.7):

Countable Initial Segments of the Degrees of Unsolvability

The problem of identifying the order types of the countable initial segments of the degrees of unsolvability was first tackled by Clifford Spector more than 20 years ago, and has since given rise to a series of papers, In this thesis a complete characterization of these order types is given by proving the following theorem: any countable upper semilattice with least element can be embedded as an initial segment of the degrees.

Degrees of Recursively Enumerable Sets Which Have No Maximal Supersets

The purpose of this paper is to present two new theorems concerning the degrees of coinfinite recursively enumerable (r.e.) sets which have no maximal supersets Let the class of all such degrees be denoted by A. Martin in [2] conjectured that there was some equality or inequality involving a′ or a″ characterizing the degrees a in A. Martin himself proved ([2, Corollary 4.1]) that a′ = 0″ is sufficient for ar r.e. degree a to be in A, and Robinson [3] announced that a′ ≥ 0″ is necessary. In this paper we improve both of these theorems by a factor of the jump, i.e., we shall show that a″ = 0″ is sufficient for an r.e. degree a to be in A , and that a″ ≥ 0″ is necessary.

A NOTE ON UNIVERSAL SETS

In this note is proved the following: THEOREM. If A x B is universal and one of A, B is r.e. then one of A, B is universal. Let a, -r be 1-argument recursive functions such that x goes to (a(x), 7(x)) is a (1-1) map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is called universal if every r.e. set is (many-one) reducible to A; A x B is called universal if the set

Uniform enumeration operations

Sacks [2] has asked whether there exists a uniform solution to Posts problem, i.e. an enumeration operation W such that d W(d) d ′ for every degree d . It is shown here that if such an operation W exists it cannot itself in a particular technical sense be uniform. In fact, the jump operation is characterized amongst such uniform enumeration operations by the condition: d W(d) for all d . In addition, it is proved that the only other uniform enumeration operations such that d ≤ W(d) for all d are those which equal the identity operation above some fixed degree.

Recursive Real Numbers

The present work was inspired by Mostowskis paper [3] in which he considers classes of sequences of real numbers, and associated sets of primitive recursive real numbers. The present treatment will differ from that of [3] in that we consider the real numbers in (0, ∞), instead of just those in (0, 1), but this will make little difference. In §1 we develop a general theory, and in §2 we show the relation of this theory to the results contained in [3], and to those of other authors.

Two Theorems on Degrees of Models of True Arithmetic

Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣ M ∣ is ω . The degree of such a model M , denoted by deg( M ), is the (Turing) degree of the atomic diagram of M . The results of this paper concern the degrees of models of N , but here in the Introduction, we shall give a brief survey of results about degrees of models of PA. Let D 0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg( M ) = d . Here are some of the more easily stated results about D 0 . (1) There is no recursive nonstandard model of PA; i.e. , 0 ∈ D 0 . This is a result of Tennenbaum [T]. (2) There exists d ∈ D 0 such that d ≤ 0 ′. This follows from the standard Henkin argument. (3) There exists d ∈ D 0 such that d 0 ′. Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem. (4) There exists d ∈ D 0 such that d ′ = 0 ′. Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4). (5) D 0 = D c = D e , where D c denotes the set of degrees of completions of PA and D e the set of degrees d such that d separates a pair of effectively inseparable r.e. sets. Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that D c is upward closed.