On sait que par le choix d’ un langage suffisamment complexe, toute structure peut admettre une élimination des quantificateurs, malheureusement cette extension du langage peut nous éloigner des phénomènes algébriques. Nous allons nous intéresser à l’ élimination des quantificateurs pour des paires de corps. Dans le cas des paires de corps algébriquement clos et des paires denses de corps réel clos, on obtient une élimination en ajoutant au langage des prédicats ayant une signification algébrique: on peut les exprimer en disant que pour deux ensembles algébriques et donnés, il existe des points du sous-corps rationnels pour et pas pour , ou qu’ un ensemble semi-algébrique donné a des points rationnels sur le sous-corps. Robinson avait déjà abordé de façon informelle le cas des paires denses de corps réel-clos (cf. [Ro 2, p. 198]). Partant du langage des paires de corps ordonnés, enrichi de symboles de relations correspondant à l’ indépendance algébrique, il proposait d’ ajouter pas à pas des fonctions de Skolem Herbrand pour faire disparaitre les quantificateurs existentiels des formules, mais sans préciser le langage obtenu. Ici nous approchons le problème différemment en explicitant dès le départ le langage utilisé.

Grâce à ces résultats nous pourrons étudier le cas des paires séparées de corps réels clos ainsi que des paires de corps valués henseliens. En élargissant la définition d’ ensemble algébrique à tous les symboles du langage, les prédicats relationnels ajoutés ont la même signification que dans le cas des paires de corps algébriquement clos.

En comparant les techniques employées ici avec celles déjà utilisées dans [K], [B], [D 1] et [L], on remarque qu’il est possible de traîter une grande partie de l’ étude (complétude, modèle complétude, élimination des quantificateurs) des paires de corps algébriquement clos, réel-clos ou henseliens en se basant sur des prolongements d’ isomorphismes entre sous-structures dénombrables où l’ une des deux est contenue dans une structure ω1-saturée.

The Kondo-Addison theorem says that every set A of reals contains a real which is implicit (for short, Imp). Any real which is Imp is constructible, and it is the main purpose of this note to show that a close examination of the usual Kondo-Addison argument allows us to compute (in terms of the ordinals associated with A by the classical representation theorem) where in the constructible hierarchy an element of A ∩ Imp must occur. This result is related to a basis theorem of Barwise and Fisher [1] and is accompanied by a strong counterexample which shows that certain tepid improvements of their theorem (and, a fortiori, of the main theorem in this note) are impossible. As a final flourish, the ‘classifying ordinals’ used to compute the locations of the singletons are themselves characterized using an idea from [1].

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion such that, in any -generic extension of the universe, there are non-isomorphic models M1 and M2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if ‘c.c.c’ is replaced by other cardinal-preserving adjectives. We also give an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings.

It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is Lévy-collapsed to ω2 the new ω2 inherits some of the large cardinal properties; e.g. if S is a stationary set of ω-limits in ω2 then for some α < ω2, S ∩ α is stationary in α. Later S. Shelah extended this to the following theorem: if a supercompact cardinal κ is Lévy-collapsed to ω2, then in the resulting model the following holds: if S ⊆ λ is a stationary set of ω-limits and cf(λ) ≥ ω2 then there is an α. < λ such that S ∩ α is stationary in α, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality μ as well, i.e. when an appropriate large cardinal is collapsed to μ++. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of ω-limits remains stationary after forcing with an ω2-closed partial order may be false in the case of μ-limits in a cardinal of the form λ+ with cf(λ) < μ, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality ≤ μ exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond μ.

We answer some questions from [4] by giving suitable examples of small Polish structures. First, we present a class of small Polish group structures without generic elements. Next, we construct a first example of a small non-zero-dimensional Polish G-group.