Our aim is to solve a quite old question on the difference between expandability and compact expandability. Toward this, we further investigate the logic of countable cofinality.

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.

In this article, unpublished methods of Solovay and Kunen are applied to describe conditions under which an uncountable regular κ can satisfy weak (κ, λ)-compactness (see 1.1(3) below), yet lie below 2μ for some μ < κ. The argumentation is in informal ZFC, and general set-theoretic notation is standard. The lower-case Greek letters κ, λ, μ, ν are reserved for cardinals in the sense of some transitive or inner model of a reasonable set theory, φ, Ψ, θ, are (arithmetizations of) formulas in some extension of the first-order language of set theory, and other lower-case Greek letters except Є are metavariables for arbitrary ordinals. If M is transitive, M ⊨ φ abbreviates 〈M, Є 〉 ⊨ φ. [2], [3], [9] and [1] provide more information about large-cardinal theory for those who wish it.

1.1. Definitions. (1) κ is inaccessible iff κ is regular and ℵκ = κ; strongly inaccessible iff κ is regular and ℶκ = κ, i.e., λ < κ for all λ < κ; weakly inaccessible iff κ is inaccessible but not strongly inaccessible.

(2) L κλ is the infinitary language with conjunctions and disjunctions of length < κ and quantification over sequences of length < λ, and PL κ the prepositional language with κ letters and conjunctions and disjunctions of length < κ.

A constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

In this note we present a simplification of the axiom schemes which were given in our earlier paper Axiom schemes for m-valued propositional calculi. Except for new concepts that are needed for our present purposes, the terminology used will be the same as that introduced in the earlier paper. Hence, we can say that in addition to listing our simplified specifications for axiom schemes, we shall indicate how our new axiom schemes may be used to establish deductive completeness.

The simplification of our axiom schemes is achieved by using only the operators ⊃ and Jk( ) with the resulting elimination of ∨ and ·. This does not constitute an actual reduction in functional complexity since the latter operators can be denned in terms of the former. For example,

However, by dispensing with ∨ and · one can greatly reduce the number of axioms without an appreciable increase in the difficulties of proving deductive completeness.

Our simplified specifications for axiom schemes will retain conditions (1) and (3) of our earlier set but (2) and (4) will be changed to (2*) and (4*) which we will now state.

(2*) Assume that an implication function (P ⊃ Q) can be defined exclusively in terms of the basic functions Fi(P1, …:, Pαi).

The statement of what follows is simplified by introducing a chain symbol defined recursively as follows:

The variable i of a chain symbol which takes values from u to v will be called the chain variable. When it is clear that i is the chain variable we will use the notation in place of . Hence, denotes (P5 ⊃ (P4 ⊃ (P3 ⊃ Q))).

We prove the undecidability of a very large class of restricted and unrestricted wreath products (Theorem 1.2), and of some skew fields of power series (Section2). Both undecidabilities are obtained by interpreting some enrichments of twisted wreath products, which are themselves proved to be undecidable (Proposition 1.1).

We consider division rings of power series in various languages:

We show (Theorem 2.8) that every power series division ring k((B)), whose field of constants k is commutative and whose ordered group of exponents is noncommutative with a convex center, is undecidable in every extension of the language of rings where the valuation and the ordered group B are definable.

For certain k and B we prove here the undecidability of the structure

where X↾k((B))xB is the restriction of the multiplication to k((B)) Χ B,and γ is a given conjugation of k((B)). This shows that we cannot hope to improve our previous result, a sort of Ax-Kochen-Ershov principle for power series division rings, which ensures that

is decidable for every decidable solvable B.

In [1] Craig introduced a proof procedure for first order classical logic called linear reasoning. In it, a proof of P ⊃ Q consists of a sequence of formulas, each of which implies the next, beginning with P and ending with Q. And one of the formulas in the sequence will be an interpolation formula for P ⊃ Q. Indeed, this was the first proof of the Craig interpolation theorem, some of whose important consequences were demonstrated in a companion paper [2]. In this paper we present systems of linear reasoning for several standard modal logics: K, T, K4, S4, D, D4, and GL. Similar systems can be constructed for several regular, nonnormal modal logics too, though we do not do so here. And just as in the classical case, interpolation theorems are easy consequences. Such theorems are well known for the logics considered here. There is a model theoretic argument in [6], an argument using Gentzen systems in [8], an argument using consistency properties in [4] and [5], and an argument using symmetric Gentzen systems in [5]. This paper presents what seems to be the first modal proof that follows Craig's original methods. We note that if the modal rules given here are dropped, a classical linear reasoning system results, which is related to, but not the same as those in [1] and [10].

Since the basic linear reasoning ideas are fully illustrated by the propositional case, we present that first, to keep the clutter down. Later we show how the techniques can generally be extended to encompass quantifiers. We do not follow [1] in making heavy use of prenex form, since it is not generally available in modal logics. Fortunately, it plays no essential role.

We show that if κ is a weakly compact cardinal, then

for any ordinals α < κ+ and μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partition

of κ × κ+ into m + μ pieces either there are A ∈ [κ]κ, B ∈ [κ]+]α and i < m with A × B ⊆ Ki or there are C ∈ [κ]κ, , and j < μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.

In [1] two interesting invariance notions were introduced: the notions of a set of godel numbers being invariant to automorphisms of the structures (ω, ·) and (ω, E) respectively. Here, · and E are defined by n · m ≃ φn (m) and nEm if and only if n Є Wm, where {φn} and {Wn} are acceptable enumerations of the partial recursive functions and r.e. sets respectively. In this paper we continue the study of the invariant sets, and especially the invariant r.e. sets, of gödel numbers.

We start off with an easy result which characterizes the Turing degrees containing invariant sets. We then take a closer look at r.e. sets invariant with respect to automorphisms of (ω,E). Using the characterization [1, Theorem 4.2] of such sets, we will derive a somewhat different characterization (which was stated, but not proved, in [1, Proposition 4.4]) and, using it as a tool for constructing invariant sets, prove that the r.e. sets invariant with respect to automorphisms of (ω, E) cannot be effectively enumerated.

We will next discuss representations of r.e. sets invariant with respect to automorphisms of (ω, ·). Although these sets do not have as nice a characterization as the r.e. sets invariant with respect to automorphisms of (ω, E) do, the techniques of [1] can still profitably be used to investigate their structure. In particular, if f is a partial recursive function whose graph is invariant with respect to automorphisms of (ω, ·), then for every a in the domain of f, there is a term t(a) built up from a and · only such that f(a) ≃ t(a). This is an analog to [1, Corollary 4.3]. We will also prove an analog to a result mentioned in the previous paragraph: the r.e. sets invariant with respect to automorphisms of (ω, ·) cannot be effectively enumerated.

We use the notation of Kripke's paper [1]. Let M = (G, K, R) be a tree structure and D a domain and η a Beth model on M. The truth conditions of the Beth semantics for ∨ and ∃ are (see [1]):

(a) η (A ∨ B, H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B, either η(A, H′) = T or η(B, H′) = T.

(b) η(∃xA(x), H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B there exists an a ∈ D such that η(A (a), H′) = T.

Suppose we change the truth definition η to η* by replacing condition (b) by the condition (b*) (well known from the Kripke interpretation):

Call this type of interpretation the new version of Beth semantics. We prove

Theorem 1. Intuitionistic predicate logic is complete for the new version of the Beth semantics.

Since Beth structures are of constant domains, and in the new version of Beth semantics the truth conditions for ∧, →, ∃, ∀, ¬ are the same as for the Kripke interpretation, we get:

Corollary 2. The fragment without disjunction of the logic CD of constant domains (i.e. with the additional schema ∀x(A ∨ B(x))→ A ∨ ∀xB(x), x not free in A) equals the fragment without disjunction of intuitionistice logic.

Cette note est une suite à mon article Théories instables, this Journal, vol. 46 (1981), pp. 513–522. Quand il en a eu pris connaissance, Saharon Shelah–et c'est remplir un devoir agréable que de lui témoigner ici ma reconnaissance pour l'intérêt qu'il porte à mes travaux–m'a fait savoir qu'il pouvait en simplifier les résultats, et a publié le Théorème 1 de ce post-scriptum, pour le cas des cohéritiers, dans les “added in proof” de son Simple unstable théories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177–203, plusieurs mois avant que ne paraisse l'article qu'il améliore. On verra ici que ce théorème se généralise sans problème aux fils spéciaux, et que son corollaire, le Théorème 2, représente un net progrès sur mon résultat original, puisqu'il est à la fois plus général, plus précis, et obtenu à moindre frais, le lemme sur les ultrafiltres devenant inutile.

Quelques rappels pour commencer: je considère une théorie complète T, sans propriété d'indépendance; cela signifie que toute suite … ai … indicernable dans l'ordre est insécable, c'est-à-dire qu'il ne peut exister de formule f(x, ȳ) et de paramètres tels que f(ai, ) soit vraie cofinalement, et fausse cofinalement, dans la suite; cela signifie encore que cette suite s est découpée par la formule f(x, ) en un nombre fini de segments S1, …, Sk, la formule prenant la même valeur de vérité dans chaque segment, et des valeurs opposées dans des segments consécutifs; ce nombre k de segments sera appelé nombre d'alternance de la formule f(x, ) sur la suite s. On remarquera que, par compacité, ce nombre d'alternance est majoré en fonction seulement de f(x, ȳ).

A commutative ring R with identity is called a local ring if R has only one maximal ideal. This is equivalent to saying that the sum of two nonunits is a non-unit. Therefore the theory of all commutative local rings is axiomatizible by a finite set of A2-sentences. A commutative local ring with identity is said to be an algebraically closed local ring if every finite system of polynomial equations and inequations in one or more variables with coefficients in R which has a solution in some commutative local extension of R already has a solution in R. Much work connected with algebraically closed structures of classes of rings has been done, for example by Cherlin [2], Macintyre [4] and Lipschitz and Saracino [3]. We want to show similar results for commutative local rings with identity. Our main results are the following:

Theorem. The theory of commutative local rings with identity has no model-companion.

The finitely generic and infinitely generic local rings are algebraically closed local rings.

Theorem. There is an A3 sentence which holds for all finitely generic local rings whose negation holds in every infinitely generic local ring.

On sait que la conjonction et la négation constituent un système de connecteurs suffisant pour formuler le calcul des propositions classiques. Un certain intêrêt s'attache à ces formulations par suite des faits suivants: – les thèses du calcul intuitioniste qui ne comprennent pas d'autres connecteurs que la conjonction et la négation sont identiques aux thèses classiques;

– Lewis ayant formulé ses calculs modaux en prenant la conjonction, la négation et la possibilité comme connecteurs primitifs (voir Lewis-Langford 1932, ch. VI) et ayant été suivi par beaucoup de chercheurs s'occupant de modalités, le calcul classique fondé sur la conjonction et la négation est contenu dans la plupart des systèmes modaux que l'on trouve dans la littérature.

Un système logistique pour le calcul classique fondé sur la conjonction et la négation a été étudié par Rosser (voir Rosser 1953, ch. 4; aussi Copi 1954, ch. 7).

Dans cet article je vais définir cinq autres systèmes logistiques fondés sur les mêmes connecteurs, démontrer leur suffisance (en utilisant les résultats publiés de Rosser et de Copi), et — sauf un cas — démontrer l'indépendance des axiomes de chacun de ces systèmes.