In this paper we shall state some interesting facts concerning non-ω-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from ω-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory.

Many papers including [4] and [5] are motivated by the conjecture that every non-ω-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory T0 of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory T0 is stable and ω-categorical, then T0 cannot be extended to a theory T which has n countable models (1 < n < ω) by adding axioms for constant symbols.

In [3], Shelah defined the cardinals κn(T) and , for each theory T and n < ω. κn(T) is the least cardinal κ without a sequence (pi)i<κ of complete n-types such that pi is a forking extension of pj for all i < j < κ. It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T. (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In this paper, we shall give equivalents of which can be easily handled.

In §1 we shall state the definitions of κn(T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property.

In §2 we shall give some conditions on κ, n, and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n. We introduce the cardinal ı(T), which is essential in computing the number of types over a set which is independent over some set, and show that ı(T) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems.

Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.

The quantifiers of standard predicate logic are interpreted as ranging over domains of individuals, and interpreted formulae beginning with a quantifier make claims to the effect that something is true of every individual, i.e. of the whole domain, or of some individuals, i.e. of part of the domain. To state that something is true of all or part of a totality seems to be the basic significance of universal and existential quantification, and this by itself does not involve a specification of the structure of the totality. This means that the notion of quantification by itself does not demand totalities of individuals, i.e. atomic totalities, as domains of quantification. Nonatomic domains, such as volumes of space, or surfaces, are equally in order. So one might say that a certain predicate applies “everywhere” or “somewhere” in such a domain. All that the concept of quantification requires is a totality which is structured in terms of a part-to-whole relation, and appropriate properties that apply to part or all of the totality. Quantification does not demand that the totality have smallest parts, or atoms. There is no conflict with the sense of universal or existential quantification if the domain is nonatomic, if every one of its parts has itself proper parts.

The most general kind of quantification theory must then deal with totalities of any kind, atomic or not. The relationships among the parts of a domain are described by the theory of Boolean algebras, which we can regard as the most general characterisation of a totality, of a domain of quantification.

In this paper I shall be concerned with this generalised theory of quantification, which encompasses nonatomic domains as well as atomic and mixed domains, i.e. totalities consisting entirely or partly of individuals.

Enumeration reducibility is a reducibility between sets of natural numbers defined as follows: A is enumeration reducible to B if there is some effective operation on enumerations which when given any enumeration of B will produce an enumeration of A. One reason for interest in this reducibility is that it presents us with a natural reducibility between partial functions whose degree structure can be seen to extend the structure of the Turing degrees of unsolvability. In [7] Friedberg and Rogers gave a precise definition of enumeration reducibility, and in [12] Rogers presented a theorem of Medvedev [10] on the existence of what Case [1] was to call quasi-minimal degrees. Myhill [11] also defined this reducibility and proved that the class of quasi-minimal degrees is of second category in the usual topology. As Gutteridge [8] has shown that there are no minimal enumeration degrees (see Cooper [3]), the quasi-minimal degrees are very much of interest in the study of the structure of the enumeration degrees. In this paper we define a jump operator on the enumeration degrees which was introduced by Cooper [4], and show that every complete enumeration degree is the jump of a quasi-minimal degree. We also extend the notion of a high Turing degree to the enumeration degrees and construct a “high” quasi-minimal enumeration degree—a result which contrasts with Cooper's result in [2] that a high Turing degree cannot be minimal. Finally, we use the Sacks' Jump Theorem to characterise the jumps of the co-r.e. enumeration degrees.