Let be a finite sequence of algebras of sets given on a set with more than pairwise disjoint sets not belonging to It was shown in [4] and [5] that in this case Let us consider, instead a finite sequence of algebras It turns out that if for each natural i ≤ l there exist no less than pairwise disjoint sets not belonging to then But if l ≥ 195 and if for each natural i ≤ l there exist no less than pairwise disjoint sets not belonging to then After consideration of finite sequences of algebras, it is natural to consider countable sequences of algebras. We obtained two essentially important theorems on a countable sequence of almost σ-algebras (the concept of almost σ-algebra was introduced in [4]).

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.

Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

Subtle cardinals were first introduced in a paper by Jensen and Kunen [JK]. They show that if κ is subtle then ◇κ holds. Subtle cardinals also play an important role in [B1], where Baumgartner proposed that certain large cardinal properties should be considered as properties of their associated normal ideals. He shows that in the case of ineffables, the ideals are particularly useful, as can be seen by the following theorem, κ is ineffable if and only if κ is subtle and Π½-indescribable and the subtle and Π½-indescribable ideals cohere, i.e. they generate a proper, normal ideal (which in fact turns out to be the ineffable ideal).

In this paper we examine properties of subtle cardinals and consider methods of forcing that destroy the property of subtlety while maintaining other properties. The following is a list of results.

1) We relativize the following two facts about subtle cardinals:

i) if κ is n-subtle then {α < κ: α is not n-subtle} is n-subtle, and

ii) if κ is (n + 1)-subtle then {α < κ: α is n-subtle} is in the (n + 1)-subtle filter to subsets of κ:

i′) if A is an n-subtle subset of κ then {α ϵ A: A ∩ α is not n-subtle} is n-subtle, and

ii′) if A is an (n + 1)-subtle subset of κ then {α ϵ A: A ∩ α is n-subtle} is (n + 1)-subtle.

2) We show that although a stationary limit of subtles is subtle, a subtle limit of subtles is not necessarily 2-subtle.

3) In §3 we use the technique of forcing to turn a subtle cardinal into a κ-Mahlo cardinal that is no longer subtle.

4) In §4 we extend the results of §3 by showing how to turn an (n + 1)-subtle cardinal into an n-subtle cardinal that is no longer (n + 1)-subtle.

If is a pseudo-variety, then a supersimple residually group is nilpotent-by-poly-.

In the study of categories whose morphisms display a behaviour similar to that of partial functions, the concept of morphism domain is, obviously, central. In this paper an operation defined on morphisms describes those properties which are related to morphisms being regarded as abstractions of partial functions. This operation allows us to characterise the morphism domains directly, and gives rise to an algebra defined by a simple set of identities. No product-like categorical structures are needed therefore. We also develop the construction of topologies together with the notion of continuous morphism, in order to test the effectiveness of this approach. It is interesting to see how much of the computational character of the morphisms is translated into continuity.

Let T be a recursively enumerable theory extending Elementary Arithmetic EA. L. D. Beklemishev proved that the Σ2 local reflection principle for T, (T), is conservative over the Σ1 local reflection principle, (T), with respect to boolean combinations of Σ1-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that Π2-sentences are not conserved for T = EA + “f is total,” where f is any nondecreasing computable function with elementary graph. We also discuss how this result generalizes to n > 0 and obtain as an application that for n > 0, is conservative over IΣn with respect to Πn+2-sentences.

A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).

Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).

Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.

The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.

In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:

(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.

(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.

(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.

(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).

In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.

Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).