It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.

We prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).

A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.

Our main result uses constructible sets and can be put in an abstract general framework.

Comparison with the standard methods which use forcing (cf. [4]) is sketched.

A countable, atomically stable structure in a finite, relational language has fewer than 2ω non-isomorphic substructures if and only if is cellular. An example shows that the finiteness of the language is necessary.

In his 1936 paper, On the Concept of Logical Consequence, Tarski introduced the celebrated definition of logical consequence: “The sentenceσ follows logically from the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentence σ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentence σ is false. Moreover, … we are concerned here with the concept of logical, i.e., formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since.

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

For a primitive recursive consistent and strong enough theory T we construct an independent statement which has some clear metamathematical meaning.

We follow [8] in asking when a set of ordinals X ⊆ α is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical Σ1 Skolem function for Kα can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

There exists a family of sets of countable ordinals such that:

(1) max Bα = α,

(2) if α ∈ Bβ then Bα ⊆ Bβ,

(3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ,

(4) there is a partition of ω1 such that for every α and every n, Bα ∩ An is finite.

Assume T is stable, small and Φ(x) is a formula of L(T). We study the impact on T⌈Φ of naming finitely many elements of a model of T. We consider the cases of T⌈Φ which is ω-stable or superstable of finite rank. In these cases we prove that if T has countable models and Q = Φ(M) is countable and atomic or saturated, then any good type in S(Q) is τ-stable. If T⌈Φ is ω-stable and (bounded, 1-based or of finite rank) with , then we prove that every good p ∈ S(Q) is τ-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories.