Dans [2], Salma et Franz Viktor Kuhlmann définissent plu-sieurs notions d'espace vectoriel valué et démontrent un principe d'Ax-Kochen pour celles de ces structures dans lesquelles la multiplication par un scalaire du corps préserve la valuation. Nous travaillons ici avec des conditions plus faibles. On va définir en premier lieu la notion d'espace vectoriel valué sur un corps K, et associer canoniquement à un tel espace une valuation sur K, notée w, compatible en un sens à préciser avec celle de l'espace vectoriel. Ceci va nous permettre de parler de (K, w)-espaces vectoriels valués pour tout corps valué (K, w) (definition 24). Le cas où (K, w) est trivialement valué correspond exactement à la condition de [2] de préservation de la valuation par la multiplication par un scalaire. On fixe un corps K. Les espaces vectoriels valués sur le corps K vont être traités comme des structures à deux sortes: la sorte de l'espace vectoriel et celle de l'espace des valuations. Le langage LE de la sorte de l'espace vectoriel contient le symbole de la somme, celui de l'élément neutre, et un symbole de fonction unaire pour chaque scalaire de K. Le langage LV de l'espace des valuations contient une relation d'ordre total, un symbole de fonction unaire pour la multiplication par chaque scalaire de K, et pour tout n ∈ ℕ*, un symbole de prédicat Rn contrôlant la cardinalité résiduelle. LV contient aussi un symbole de fonction unaire s qui sera la fonction successeur quand elle est définie et l'identité sinon.

Let Π be a complete théorie of pseudo-finite fields. In this article we prove that, in the langage of fields to which we add a unary predicate for a substructure, the theory of non trivial elementary pairs of models of Π has completions, that is, the maximum that could exist.

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.

The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

In most published systems of modal logic only one or two modal operators occur as undefined constants alongside the primitives of non-modal logic. But in a single informal argument many non-interdefinable modal operators may occur. E.g. in philosophy we may have ‘it is logically true that’, ‘it is analytic that’, and ‘it is physically necessary that’; and in jurisprudence or private international law we may have statements about what it is obligatory or not obligatory to do under several different systems of rules. Moreover even if the contexts of such operators are all referentially opaque, in Quine's sense [1] of being exempt from the substitutivity of truth-functional biconditionals, some are, as it were, opaquer than others, in being exempt from the substitutivity of certain non-truth-functional biconditionals as well.

We characterize some large cardinal properties, such as μ-measurability and P2(κ)-measurability, in terms of ultrafilters, and then explore the Rudin-Keisler (RK) relations between these ultrafilters and supercompact measures on Pκ(2κ). This leads to the characterization of 2κ-supercompactness in terms of a measure on measure sequences, and also to the study of a certain natural subset, Fullκ, of Pκ(2κ) whose elements code measures on cardinals less than κ. The hypothesis that Fullκ is stationary (a weaker assumption than 2κ-supercompactness) is equivalent to a higher order Löwenheim-Skolem property, and settles a question about directed versus chain-type unions on Pκλ.

We denote by KP_ the fragment of set-theory containing the axioms of extensionality, pairing, union and foundation as well as the schemas of ∆0-comprehension and ∆0-collection, that is: Kripke-Platek set-theory (KP) with the axiom of foundation in place of the ∈-induction schema. The theory KP is obtained by adding to KP_ the schema of ∈-induction

Using ∈-induction it is possible to prove the existence of the transi tive closure without appealing to the axiom of infinity (see, e.g., [1]). Vice versa, when a theory proves the existence of the transitive closure, some induction is immediately ensured (by foundation and comprehension). This is not true in general: e.g., the whole of Zermelo-Fraenkel set-theory without the axiom of infinity does not prove ∈-induction (in fact, it does not prove the existence of the transitive closure; see, e.g., [3]). Open-induction is the schema of ∈-induction restricted to open formulas. We prove the following theorem.

KP_ proves open-induction.

We reason in a fixed but arbitrary model of KP_ whom we refer to as the model. The language is extended with a name for every set in the model. We call this constants parameters. Let φ(x) be a satisfiable open-formula possibly depending on parameters and with no free variable but x. We show that φ(x) is satisfied by an ∈-minimal set, that is, a set a such that φ(a) and (∀x ∈ a) ¬φ(x). We assume that no ordinal satisfies φ(x), otherwise the existence of a ∈-minimal set follows from foundation and comprehension.

There are two main axiomatic extensions of Zermelo-Fraenkel set theory without the axiom of choice, that associated with the axiom of determinateness, and that associated with infinite exponent partition relations. Initially, the axiom of determinateness, henceforth AD, was the sole tool available. Using it, set theorists in the late 1960s produced many remarkable results in pure set theory (e.g. the measurability of ℵ1) as well as in projective set theory (e.g. reduction principles for ). Infinite exponent partition relations were first studied successfully soon after these early consequences of AD. They too produced measurable cardinals and not only were the constructions here easier than those from AD—the results gave a far clearer picture of the measures involved than had been offered by AD. In general, the techniques offered by infinite exponent partition relations became so attractive that a great deal of the subsequent work from AD involved an initial derivation from AD of the appropriate infinite exponent partition relation and then the derivation from the partition relation of the desired result.

Since the early 1970s work on choiceless extensions of ZF + DC has split mainly between AD and its applications to projective set theory, and infinite exponent partition relations and their applications to pure set theory. There has certainly been a fair amount of interplay between the two, but for the most part the theories have been pursued independently.

Unlike AD, infinite exponent partition relations have shown themselves amenable to nontrivial forcing arguments. For example, Spector has constructed models for interesting partition relations, consequences of AD, in which AD is false. Thus AD is a strictly stronger assumption than are various infinite exponent partition relations. Furthermore, Woodin has recently proved the consistency of infinite exponent partition relations relative to assumptions consistent with the axiom of choice, in particular, relative to the existence of a supercompact cardinal. The notion of doing this for AD is not even considered.

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.