It is shown that the nonrecursive predecessors of a 1-generic degree < 0′ are all 1-generic. As a corollary, it is shown that the 1-generic degrees are not densely ordered.

There are several conceptions of truth, such as the classical correspondence conception, the coherence conception and the pragmatic conception. The classical correspondence conception, or Aristotelian conception, received a mathematical treatment in the hands of Tarski (cf. Tarski [1935] and [1944]), which was the starting point of a great progress in logic and in mathematics. In effect, Tarski's semantic ideas, especially his semantic characterization of truth, have exerted a major influence on various disciplines, besides logic and mathematics; for instance, linguistics, the philosophy of science, and the theory of knowledge.

The importance of the Tarskian investigations derives, among other things, from the fact that they constitute a mathematical, formal mark to serve as a reference for the philosophical (informal) conceptions of truth. Today the philosopher knows that the classical conception can be developed and that it is free from paradoxes and other difficulties, if certain precautions are taken.

We believe that is not an exaggeration if we assert that Tarski's theory should be considered as one of the greatest accomplishments of logic and mathematics of our time, an accomplishment which is also of extraordinary relevance to philosophy, as we have already remarked.

In this paper we show that the pragmatic conception of truth, at least in one of its possible interpretations, has also a mathematical formulation, similar in spirit to that given by Tarski to the classical correspondence conception.

The complete diagram of a structure , denoted by Dc(), is the set of all sentences true in the structure (, a)a∈. A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc(), if θ is consistent with Dc(), then there is a relation P on such that (see[1]).

Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.)

The proof of the main result requires two lemmas.

It is shown that Friedman's problem, whether there exists a proper extension of first order logic satisfying the compactness and interpolation theorems, has extremely simple positive solutions if one considers extensions by generalized (finitary) propositional connectives. This does not solve, however, the problem of whether such extensions exist which are also closed under relativization of formulas.

Given an abstract logic , generated by a set of quantifiers Qi, one can construct for each type τ a topological space Sτ, exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set is an inverse topological system whose bonding mappings are naturally determined by the reduct operation on structures. We relate the compactness of to the topological properties of ST. For example, if I is countable then is compact iff for every τ each clopen subset of Sτ is of finite type and Sτ, is homeomorphic to limST, where T is the set of finite subtypes of τ. We finally apply our results to concrete logics.

Le prophète des Crétois, nous le savions depuis longtemps, n'était qu'un menteur; pourtant, à l'aube de ce siècle, quand nous avons voulu traduire nos mathématiques en termes d'ensembles, notre conscience s'est troublée de la découverte de paradoxes.

“Si R est une relation binaire entre éléments d'un ensemble (sic) E, il n'existe pas d'élément a de E, tel que pour tout x de E (a, x) satisfasse R si et seulement si (x, x) ne satisfait pas R.” Voilà un résultat bien banal! Quel est cet assaisonement fondationnel qui a le pouvoir de lui donner une saveur si troublante? C'est, sans doute, qu'il manifeste de façon brutale la contradiction d'un système d'axiomes auquel nous sommes viscéralement attachés, la “théorie des ensembles”, composée de l'axiome d'extensionalité (sans influence sur les paradoxes), et du schéma de compréhension (car nous avons d'excellentes raisons de vouloir rester dans le premier ordre, et d'éviter, à l'encontre de Zermelo, un axiome du second ordre).

Nous croyons en premier lieu que la théorie des ensembles est vraie. C'est un peu délicat de préciser ce qu'on entend par là: quand on formalise l'arithmétique, il n'y a pas de confusion possible sur la structure de référence qu'on veut décrire; mais le modèle naturel, le “modèle standard” de la théorie des ensembles, le connait-on vraiment? L'auteur de ces lignes est incapable d'en décider: cela ne l'empêchera pas de faire appel à l'intuition ensembliste la plus débridée, ce qui montre qu'il n'est pas rebuté par l'incohérence plus que par la contradiction!