THE LINDENBAUM ALGEBRA OF THE THEORY OF THE CLASS OF ALL FINITE MODELS

In this paper, we investigate the Lindenbaum algebra ℒ(T fin ) of the theory T fin = Th (M fin ) of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra (ℒ(T fin )/ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ(T fin ); moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra (ℒ(T fin )/ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature.

The relation of recursive isomorphism for countable structures

It is shown that the relations of recursive isomorphism on countable trees, groups, Boolean algebras, fields and total orderings are universal countable Borel equivalence relations, thus providing a countable analogue of the Borel completeness of the isomorphism relations on these same classes.

Definability of r. e. sets in a class of recursion theoretic structures

It is known [4, Theorem 11-X(b)] that there is only one acceptable universal function up to recursive isomorphism. It follows from this that sets definable in terms of a universal function alone are specified uniquely up to recursive isomorphism. (An example is the set K, which consists of all n such that {n}(n) is defined, where λn, m{n}(m) is an acceptable universal function.) Many of the interesting sets constructed and studied by recursion theorists, however, have definitions which involve additional notions, such as a specific enumeration of the graph of a universal function. In particular, many of these definitions make use of the interplay between the purely number-theoretic properties of indices of partial recursive functions and their purely recursion-theoretic properties.This paper concerns r.e. sets that can be defined using only a universal function and some purely number-theoretic concepts. In particular, we would like to know when certain recursion-theoretic properties of r.e. sets definable in this way are independent of the particular choice of universal function (equivalently, independent of the particular way in which godel numbers are identified with natural numbers).We will first develop a suitable model-theoretic framework for discussing this question. This will enable us to classify the formulas defining r.e. sets by their logical complexity. (We use the number of alternations of quantifiers in the prenex form of a formula as a measure of logical complexity.) We will then be able to examine the question at each level.This work is an approach to the question of when the recursion-theoretic properties of an r.e. set are independent of the particular parameters used in its construction. As such, it does not apply directly to the construction techniques most commonly used at this time for defining particular r.e. sets, e.g., the priority method. A more direct attack on this question for these techniques is represented by such works as [3] and [5]. However, the present work should be of independent interest to the logician interested in recursion theory.

Recursive properties of isomorphism types

For Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.

Recursive isomorphism types of recursive Boolean algebras

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1 → B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N − is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

Recursive Boolean algebras with recursive atoms

A Boolean algebra (henceforth abbreviated B.A.) is said to be recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Let denote the set of atoms of and denote the ideal generated by the atoms of . Given recursive B.A.s and , we write ≈ if is isomorphic to and ≈r if is recursively isomorphic to , i.e., if there is a partial recursive isomorphism from onto .Recursive B.A.s have been studied by several authors including Ershov [2], Fiener [3], [4], Goncharov [5], [6], [7], LaRoche [8], Nurtazin [7], and the author [10], [11]. This paper continues a study of the recursion theoretic relationships among , , and the recursive isomorphism type of a recursive B.A. we started in [11]. We refer the reader to [11] for any unexplained notation and definitions. In [11], we were mainly concerned with the possible recursion theoretic properties of the set of atoms in recursive B.A.s. We found that even if we insist that be recursive, there is considerable freedom for the properties of . For example, we showed that if is a recursive B.A. such that is recursive and is infinite, then (i) there exists a recursive B.A. such that and both and are recursive and (ii) for any nonzero r.e. degree δ, there exist recursive B.A.s , , … such that for each i, is of degree δ, is recursive, is immune if i is even and is not immune if i is odd, and no two B.A.s in the sequence are recursively isomorphic.

Vaught's theorem recursively revisited

In this paper we investigate the relationship between the number of countable and decidable models of a complete theory. The number of decidable models will be determined in two ways, in §1 with respect to abstract isomorphism type, and in §2 with respect to recursive isomorphism type.Definition 1. A complete theory is (α, β) if the number of countable models of T, up to abstract isomorphism, is β, and similarly the number of decidable models of T is α.Definition 2. A model is ω-decidable if ∣∣= ω and for an effective listing {θn∣n < ω} of all sentences in the language of Th() augmented by new constant symbols i*, i < ω, {n ∣〈, i〉i<ω ⊨ θn} is recursive, where i interprets i* (in these terms, is decidable if is abstractly isomorphic to an ω-decidable model).Definition 3. A complete theory is (α, β)r if it is (γ, β) for some γ and it has exactly αω-decidable models up to recursive isomorphism.Specifically we will show in §1 that there is a (2, ω) theory, and in §2 that although there is a (2, 2ω) theory, there is no (2, β)r theory for any β, β < 2ω.