Our principal result is that there exist two incomparable recursively enumerable degrees whose greatest lower bound in the upper semilattice of degrees is 0. This was conjectured by Sacks [5]. As a secondary result, we prove that on the other hand there exists a recursively enumerable degree a < 0(1) such that for no non-zero recursively enumerable degree b is 0 the greatest lower bound of a and b.

The proof of the main theorem involves a method that we have developed elsewhere [8] to deal with situations in which a partial recursive functional may interfere infinitely often with an opposed requirement of lower priority.

We study certain first and second order theories which are semantically defined as the sets of all sentences true in certain given structures. Let be a structure where A is a non-empty set, λ is an ordinal, and Pα is an n(α)-ary relation or function4 on A. With we associate a language L appropriate for which may be a first or higher order calculus. L has an n(α)-place predicate or function constant P for each α < λ. We shall study three types of languages: (1) first-order calculi with equality; (2) second-order monadic calculi which contain monadic predicate (set) variables ranging over subsets of A; (3) restricted (weak) second-order calculi which contain monadic predicate variables ranging over finite subsets of A.

Let Nk denote the set of words over the alphabet Σk = {1, …, k}. Nk contains the null word which is denoted λ. We consider decision problems for various first-order interpreted predicate languages in which the variables range over Nk (k ≧ 2). Our main result is that there is no decision procedure for truth in the interpreted language which has the subword relation as its only non-logical primitive. This, together with known results summarized in Section 4, settles the decision problem for any language constructed on the basis of the relations and functions listed below.

This paper is a sequel to [7], and the terminology of [7] is largely presupposed here. In [7], the algebraic methods of McKinsey-Tarski were employed and extended to yield semantic results of a Kripke kind for a class of relatively weak modal logics, the strongest of which was the Feysvon Wright system T. Deontic versions of both T and E2, called T(D) and D2, and even weaker systems, were handled. The main aim of the present paper is to extend these results to stronger systems of modal logic. Thus the Lewis systems S2–S5, the Brouwersche system B of Kripke [4], the systems E3–E5 of [5], and Łukasiewicz's modal logic, as well as certain new systems, are considered.

Certain modifications of the method of [7] have proved convenient. Thus in Section I, some further results concerning model structures are proved in order that the relationship between S2 and E2, S3 and E3, can be properly stated; in particular, the notions of a refined and connected model structure play a pervasive role throughout.

A Turing Machine (TM) is an abstract, synchronous, deterministic computer with a finite number of internal states. It operates on the set of infinite words, or tapes, over some finite alphabet, scanning exactly one symbol of the tape at a time. (Only a 2-symbol alphabet, consisting of “0” and “|”, will be considered here, and the scanned symbol of a tape will be distinguished by an underscore.) depending upon its internal state and the symbol under scan, it can perform one or more of the following operations: replace the scanned symbol with a new symbol, focus its attention on an adjacent square, and transfer control to a new state.

Plusieurs auteurs ont déjà considéré l'ultraproduit et l'ultrapuissance en prenant certaines, et non forcément toutes les fonctions qui, à chaque indice i, associent un élément de la base de la relation de rang i (voir par exemple [6]). La généralisation ici considérée, appelée ultraproduit complet (voir §3) est définie à l'aide des isomorphismes locaux. On y retrouve l'essentiel de la théorie de l'ultraproduit classique avec évidemment des exemples plus variés.