In this paper we deal with infinitary languages which allow arbitrary infinite disjunctions and conjunctions, but finite strings of quantifiers only. Furthermore, we shall assume that the primitive predicate symbols are finitary. (Cf. [K] and §2 for further information and unexplained notations.)

Our main result shows that every “reasonable” language of this type is, in a certain sense, reducible to one of the same type which allows countable disjunctions and conjunctions only. More precisely if we let θ be the first measurable cardinal and μ ≤ θ, we have the following result partially announced in [R1]:

Theorem. Every theory in a Lμω language is equivalent to some theory in a Lω1ω language in the sense that atomic formulas of one language can be mapped into formulas of the other in such a way that, for every set, these maps establish a bijective correspondence between their models having that set as a universe.

The paper is divided into three sections. In the first, we derive the main result from a theorem of Sikorski on ω-homomorphisms of μ-complete boolean algebras. In the second, we give a refined version of Sikorski's theorem from which cardinality bounds can be given for the Lω1ω theories obtained by our reduction. In the last, we use a Skolem ultrapower construction to give examples for which these bounds are actually attained. As a corollary, we obtain the converse of our main result.

We show that in the models of u < ∂ from [14] there are infinitely many near-coherence classes of ultrafilters, thus answering Banakh's and Blass' Question 30 of [3] negatively. By an unpublished result of Canjar, there are at least two classes in these models.

ℵ2. One of the first examples of the forcing method is cardinal collapsing (A. Levy, see [5]), for example, collapsing ℵ2 to ℵ1: the poset P is the collection of all countable functions from a countable ordinal into ℵ2. As is well known, in VP, , and because P is closed under union of countable chains, remains a cardinal and, in fact, no new countable sets are added. But to prove that remains a cardinal we need to conclude ∣P∣ ≤ ℵ2 and hence ℵ3 is not collapsed. If it has been observed that is actually collapsed in VP. Hence the following theorem, which makes no assumptions on the continuum, is relevant.

1. Theorem. There is a poset R such that in VR ℵ2 becomes of cardinality ℵ1, but ℵ1 and the cardinals above ℵ2 are not collapsed.

Before one can construct scales of minimal complexity in the Real Core Model, K(ℝ), one needs to develop the fine-structure theory of K (ℝ). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(ℝ). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(ℝ)and the strength of the theory ZF + AD + ┐DCℝ.

Introduction. In 1937 E. Čech and M.H. Stone, independently, introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 23]. In the introduction of [8] the non-constructive character of this result is so described: “It must be emphasized that β(S) [the Stone-Čech compactification of S] may be defined only formally (not constructively) since it exists only in virtue of Zermelo's theorem”.

By replacing topological spaces with locales, Banaschewski and Mulvey [4, 5, 6], and Johnstone [14] obtained choice-free intuitionistic proofs of Stone-Čech compactification. Although valid in any topos, these localic constructions rely—essentially, as is to be demonstrated—on highly impredicative principles, and thus cannot be considered as constructive in the sense of the main systems for constructive mathematics, such as Martin-Löf's constructive type theory and Aczel's constructive set theory.

In [10] I characterized the locales of which the Stone-Čech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of Dependent Choice.

We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.

In this paper we show that certain notions of consistency currently in use in illative combinatory logic are not always necessary nor always sufficient for the acceptability of the system.

The three notions of consistency we will consider are:

(A) Not all terms of the system are assertible.

(B) Not all propositions of the system are assertible.

(C) The system is Q-consistent, i.e. if Q (for equality) is in or is added to the system as a primitive with axioms such as

Post defined a system of propositional calculus to be inconsistent if all propositions were assertible. In a more general system where there may be terms which are not propositions (although all propositions are terms), we need to distinguish notions (A) and (B) as two possible consistency notions “in the sense of Post”.

In [3] Curry and Feys assume that Q is in the system defined as λxλy.Ξ(C*x)(C*y) or as a primitive, but Seldin in [5] also talks about Q-consistency of a system (F22) to which Q has to be added. We therefore consider the definition of Q-consistency that includes all three possibilities.

In [3] Curry and Feys state that Q-consistency is essential for acceptability. This is because there the purpose was to have an assertional system of illative combinatory logic in which combinatory equality was represented by the assertion of QXY. However if Q is to represent some other form of equality, Q-consistency may become inessential for acceptability or even incompatible with it.

Cette note est consacrée à quelques compléments à mon article [6], que le lecteur est supposé avoir sous les yeux, et qui traite de la propriété de recouvrement fini, de Keisler. Je rappelle seulement ici que:

a) La définition originale est la suivante: une théorie complète T n'a pas la p.r.f. si à toute formule f(x, ȳ) de son langage est associé un entier n, de sorte que tout ensemble (fini!) {f(x, ā1), …, f(x, ās)}, où les āi; sont des paramètres extraits d'un modèle M de T, soit consistant dès que ses parties à n éléments le sont. Une théorie sans p.r.f. satisfait done un renforcement du théorème de compacité; elle est en ce sens plus simple qu'une théorie avec p.r.f.

b) T n'a pas la p.r.f. si et seulement si elle est stable (i.e. tout type sur un modèle M de T est définissable; à toute formule f(x, ȳ) est associée une formule g(ȳ, ā) à paramètres dans M, telle que les de M pour lesquels f(x, ) est dans le type en question soient précisément ceux qui vérifient g(ȳ, ā)), et si en outre on peut exprimer par une formule sur le paramètre ā que g(ȳ, ā) définit un f-type consistant: à f1(x, ȳ), …, fs(x, ȳ), g1(ȳ, ), …, gs(ȳ, ) est associé une formule h(), telle que pour tout ā de tout modèle M de T, ā satisfasse h si et seulement si l'ensemble de tous les fi(x, ), où est dans M et satisfait gi(ȳ, ā), est consistant.

One of the objectives of mathematics is to construct suitable models for practical or theoretical phenomena and to explore the mathematical richness of such models. This enables other scientists to obtain a better understanding of such phenomena. As an example we will mention the real line and related structures. The line can be used profitably in the study of discrete phenomena like population growth, chemical reactions, etc.

Today's version of the real line is a topological completion of the rational numbers. This is so because then mathematicians have been able to work out a powerful analysis of the line. By using the real line to construct models for finitary phenomena we are more able to study those phenomena than we would have been sticking only to true-to-nature but finite structures.

So we may say that the line is a mathematical model for certain finite structures. This motivates us to seek natural models for other types of finite structures, and it is natural to look for models that in some sense are complete.

In this paper our starting point will be finite systems of finite operators. For the sake of simplicity we assume that they all are operators of one variable and that all the values are natural numbers. There is a natural extension of the systems such that they accept several variables and give finite operators as values, but the notational complexity will then obscure the idea of the construction.

Generalized n-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).

In a simplified world, every generalized n-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a dramatic way.

Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.

In this article we show that the same phenomenon occurs for the generalized n-gons if n ≥ 3 is an odd integer. For each such n we construct many nonisomorphic generalized n-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] for n = 3 carry over to an arbitrary positive odd integer n (which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphs K* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property.

A minimal formula is a formula which is minimal in provable formulas with respect to the substitution relation. This paper shows the following: (1) A β-normal proof of a minimal formula of depth 2 is unique in NJ. (2) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NJ. (3) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NK.

We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe Imp, which they introduced in [5]. Specifically, we show that it is relatively consistent with ZFC that $$Imp = \neg {\rm{CH}}$$, that $Imp \ne {\rm{HOD}}$, and that $$Imp \models V \ne Imp$$, or in other words, that $\left( {Imp} \right)^{Imp} \ne Imp$.

We generalize tools and results from first order stable theories to groups inside a simple stable strongly homogeneous model.