Mostowski [11] shows that if a structure has a decidable theory, then its weak direct power has one as well; his proof however never produces decision procedures which are elementary recursive.

Some very general results are obtained here about the nature of the weak direct power of a structure, which in most cases lead to elementary recursive decision procedures for weak direct powers of structures which themselves have elementary recursive procedures.

In particular, it is shown that 〈N*, +〉, the weak direct power of 〈N, +〉, can be decided in space

for some constant c. As corollaries, the same upper bound is obtained for the theory of the structure 〈N +, ·〉 of positive integers under multiplication, and for the theory of finite abelian groups. Fischer and Rabin [7] have shown that the theory of 〈N,* +〉 requires time

even on nondeterministic Turing machines.

In [2] we described an arithmetic theory of constructions (ATC) and showed that first-order intuitionistic arithmetic (HA) could be interpreted in it. In [3] we went on to show that the interpretation of HA in ATC is faithful. The purpose of the present paper is to apply these ideas to intuitionistic arithmetic in all finite types. Tait has shown [6] that a conservative extension of HA is obtained by adding the Gödel functionals with intuitionistic logic and intensional identity in all finite types. Below we show that this extension remains conservative on the addition of certain axioms of choice which are evident on the intended interpretation of the intuitionistic logical connectives. This theorem (Corollary 6.2 below) was first obtained by a more complicated argument in our dissertation [1]. Some of its implications are discussed in Goodman and Myhill [4].

We assume that the reader is familiar with [2] and [3].

The main result of this paper is the fact that for T a complete extension of either P or ZF + V = L, with no new symbols, there is a type which is omitted in a model of T of power ℵ1 but which is realized in all models of higher power. Therefore, the “Hanf number” for omitting types over T is greater than ℵ2.

The present section contains some basic definitions and a discussion of related results. §1 discusses generic relations on models of P and ZFC, and describes a special forcing technique due to Addison. In §2, this technique is used to obtain the main result.

The use of forcing is not essential. Those who do not wish to read a forcing argument may skip §1. At the point in §2 where forcing is used, a proof without forcing is sketched. This proof was obtained only after a careful study of the forcing argument. The forcing argument is given because it is simple, and because the technique of forcing makes intuitively clear why this and other similar results should be true.

All theories in the paper are assumed to be countable and to have infinite models. Let T be a complete theory in a language L. The Hanf number for omitting types over T, denoted by H(T), is the first infinite cardinal κ such that for all L-types, Σ, if T has models omitting Σ in all infinite powers less than κ, then T has models omitting Σ in all infinite powers.

Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical.

Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive.

Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p-adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy.

In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p-adic fields. We shall outline a new treatment of the p-adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p-adically closed fields.

We want to describe the definable subsets of p-adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K). In particular, if X is infinite, X has nonempty interior.

Now, there is an analogous question for p-adically closed fields. If K is p-adically closed, what are the definable subsets of K? To the best of our knowledge, this question has not been answered until now.

What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p-adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

Priority arguments have been used to produce r.e. sets whose complements have various properties (e.g., maximal sets, hyperhypersimple sets, etc.). In this paper we use the priority method in the context of certain recursively presented models (e.g., dense unbordered linear orderings, countable atomless Boolean algebras, infinite dimensional vector spaces over a finite field) to produce hypersimple subsets of these models whose complements are closed under algebraic operations and/or whose complements are elementarily isomorphic to certain algebraic substructures. For example, it follows from the results of §3 that there are hypersimple subsets of a countable atomless Boolean algebra (of an infinite dimensional vector space over a finite field) in every nonzero r.e. degree whose complement is a subalgebra (subspace). Also, it follows from the same result that, given any subset P of the rationals Q, there is a hypersimple subset of Q whose complement is order isomorphic to P.

A model M of a theory T in a countable language L is said to be recursively presented if the universe ∣M∣ of M is an initial segment of the natural numbers and the satisfaction relation “a satisfies Φ in M” is recursive (where a is a finite sequence of elements of ∣M∣ and Φ is a formula in L).

We are concerned in this paper with infinite binary sequences which are noncomplex in the sense that their minimal-program complexity (i.e., the lengths of shortest programs for computing their finite initial segments), as a function of the lengths of their initial segments, grows arbitrarily (in an effective sense) slowly. Indeed, the existence of such sequences which are also nonrecursive raises some interesting questions concerning the notion of computability itself. Our first task is to give a characterization of these sequences which does not directly involve the notions of complexity theory. Although these characterizations do involve the primitives of recursive function theory, they do not involve the types of properties which one usually encounters there. While a closer connection is still hoped for, the lack of one is not entirely unexpected. For example, except for the trivial case of degree 0, there is no general correlation between program complexity and degrees of unsolvability. The reason, roughly, is this: even though the inequality deg (f) ≤ deg (g) expresses the relative information between f and g (in the sense that if one knows g then one can compute f), it is a qualitative sort of information. The minimal-program complexity is used as a measure of algorithmic information content and as such is quantitative.

The second task of this paper is to present an example of one of these sequences which to its credit possesses a fairly long list of interesting properties, not the least attractive of which is that, though nonrecursive, it is very simple to describe. In fact this sequence has occurred previously in the literature. It is the characteristic sequence of the set of busy beaver numbers first studied by T. Rado [13]. Of particular interest to the discussion in the last section of this paper concerning the notion of computability is the fact that all the initial segments of this sequence are computable by arbitrarily short programs each of which is defined on all inputs and runs very quickly.

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by

(i) enlarging our language so as to allow expressions of infinite length, and

(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.

In March 1973, W. Veldman [1] discovered that, by a slight modification of a Kripke-model, it was possible to give an intuitionistic proof of the completeness-theorem for the intuitionistic predicate calculus (IPC) with respect to modified Kripke models. The modification was the following: Let f represent absurdity, then we allow the possibility that and we agree that, for all sentences ϕ, , if . Just one modified Kripke model is constructed such that validity in implies derivability in IPC. While usually one thinks of as some subset of ⋃ n Nat n and of as the discrete natural ordering in ⋃ n Nat n, in Veldman's model , is a spread and , where Γα and Γβ are sets of sentences associated with α, resp. β, is a nondiscrete ordering.

In the completeness-proofs, both for Beth and for Kripke models that we present here, we consider only models over ⋃n Nat n, with the natural discrete ordering and we need validity in all models, not just in one, to get derivability in IPC. Also we have to modify the definition of a model in a somewhat different way than Veldman did. We agree that if ∨s[M⊨s f], then M⊨sϕ for each s ∈ ⋃n Nat n and for each sentence ϕ.

One can view a single model of the type constructed in [1] as the result of throwing together all the models of (the type constructed in) this paper into one big model, which has the somewhat strange properties mentioned above.

The existence of an ω2-Souslin tree will be proved (Theorem 2.2 or §3) from the Generalized Continuum Hypothesis (GCH) plus Jensen's combinatorial principle □ω1. Thus, it follows from Jensen's 1.4(2) that the consistency of the formal theory T given by ZFC + GCH + “ω2-Souslin Hypothesis” implies the consistency of ZFC + “there exists a Mahlo cardinal.” So one does not hope to prove the consistency of this T relative to the consistency of ZFC + “there is an inaccessible cardinal, hence there are transitive models of ZFC.”

Silver [5, Theorem 5.8] has shown that the consistency of ZFC + “there is a weakly compact cardinal” implies the consistency of ZFC + not GCH + “there is no ω2-Aronszajn tree, hence no ω2-Souslin tree”; this is one reason why we deal with GCH here. Jensen has shown that the consistency of ZFC implies the consistency of ZFC +GCH + “ω1-Souslin Hypothesis.”

In the preliminary §1, we state some definitions and known results about trees and some of Jensen's combinatorial principles, including □ and ◇*(E).

Our main Lemma 2.1 states (a fortiori) that GCH implies ◇* at ω-cofinal elements of ω2 (i.e., in our notation, ◇*(E(ω) ∩ ω2)). From Lemma 2.1 and the known facts of §1, it is proved (2.5) that if □ k , the cofinality cf(k)> ω, and GCH, then there is a k +-Souslin tree. For k = ω1, this implies the result mentioned above for ω2-Souslin trees.

With quantifier elimination and restriction of language to a binary relation symbol and constant symbols it is shown that countable complete theories having three isomorphism types of countable models are “essentially” the Ehrenfeucht example [4, §6].

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.

Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.

Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.

If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg ( A ) = {dg(A): A ∈ A ). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg ( M ), dg ( HHS ), and dg ( DS ) are all equal to the set H of RE degrees a such that a ′ = 0″.

Let M * be the class of coinfinite RE sets having no superset in M ; and define HHS * and DS * similarly. Martin showed that dg ( DS *) = H. In [2], Lachlan showed (among other things) that dg ( M *)⊆K, where K is the set of RE degrees a such that a ″ > 0″. We will show that K ⊆ dg ( HHS *). Since maximal sets are hyperhypersimple, this gives dg ( M *) = dg ( HHS *) = K.

These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.