Une bijection p de M 2 dans M est dite sans cycles, ou encore localement libre, si pour tout terme t(x 1,…,xn ) formé à partir de p, et qui n'est pas une simple variable, on a, pour tous a 1,…,an de M, t(a 1,… an ) ≠ a 1; toutes ces “fonctions-paires” sont élémentairement équivalentes, à condition, bien sûr, que l'ensemble M soit non vide, et leur théorie (parfois traduite dans un autre langage) a été considérée par divers auteurs.

Ainsi, A. Mal'cev [Mal'cev 1962] a décrit toutes les théories d'algèbre localement libres, associées à un langage fonctionnel arbitraire; O. Belegradek [Belegradek] a étudié dans le détail leurs propriétés de stabilité; F. Lucas [Lucas 1978] avait également considéré des fonctions-paires localement libres; c'est encore une fonction-paire qui a été utilisée par J. F. Pabion [Pabion 1982], pour ω1-saturer, mais pas davantage, les modèles de l'Arithmétique; il a été remarqué, dans [Poizat 1984], combien il était utile pour cela de disposer d'une théorie non superstable sans propriété de recouvrement fini; dans un tout autre contexte, et pour de tout autres raisons, cette théorie de fonctions-paires a également été utilisée dans [Poizat 1986]; tous ces emplois supposent qu'on y maîtrise bien l'élimination des quanteurs.

Le rêve secret de tout logicien, c’est de prouver un résultat mathématique significatif avec des moyens de fortune; ce rêve se réalise parfois de manière quelque peu biaisée, le théorème obtenu n’étant qu’une version trop simplifiée, ou bien trop adaptée aux besoins de la logique, pour convaincre un mathématicien normal. C’est pour cela que j’annonce d’emblée la couleur, et que je précise les règles du jeu: la version du théorème de Borel-Tits que je vais montrer, concernant les groupes algébriques simples sur un corps de base algébriquement clos, sera considérée comme pratiquement évidente par un géomètre; mais c’est, à mon avis, la seule qui ait un intérêt pour un théoricien des modèles.

Quand on entreprend ainsi de redémontrer une version simple d’un résultat par ailleurs bien connu, le seul intérêt est dans la méthode: ce que je veux, ici, c’est présenter une preuve qui n’utilise aucune information, ou presque, sur la structure algébrique de ces groupes; il est même souhaitable d’oublier qu’il s’agit de groupes linéaires! Elle repose sur des résultats généraux concernant les groupes de rang de Morley fini, dus à divers auteurs, dont le principal, Boris Iosifovič Zil′ber, a déjà fait une tentative similaire [Zil′ber 1984]; je poursuis ici cette tentative, mais en me limitant à des arguments encore moins spécifiques au contexte de la géométrie.

Si je fais ainsi, ce n’est pas pour donner l’impression que l’unique ambition de la théorie des modèles est de montrer des résultats triviaux par des méthodes triviales.

In Hilbert and Ackermann [2] there is a simple proof of the consistency of first order predicate logic by reducing it to propositional logic. Intuitively, the proof is based on interpreting predicate logic in a domain with only one element. Tarski [7] and Gentzen [1] have extended this method to simple type theory by starting with an individual domain consisting of a single element and then interpreting a higher type by the set of truth valued functions on the previous type.

I will use the method of Hilbert and Ackermann on Martin-Löf's type theory without universes to show that ¬Eq(A, a, b) cannot be derived without universes for any type A and any objects a and b of type A. In particular, this proves the conjecture in Martin-Löf [5] that Peano's fourth axiom (∀x ϵ N)¬ Eq(N, 0, succ(x)) cannot be proved in type theory without universes. If by consistency we mean that there is no closed term of the empty type, then the construction will also give a consistency proof by finitary methods of Martin-Löf's type theory without universes. So, without universes, the logic obtained by interpreting propositions as types is surprisingly weak. This is in sharp contrast with type theory as a computational system, since, for instance, the proof that every object of a type can be computed to normal form cannot be formalized in first order arithmetic.

We consider countably additive, nonnegative, extended real-valued measures which vanish on singletons. Such a measure is universal on a set X iff it is defined on all subsets of X and is semiregular iff every set of positive measure contains a subset of positive finite measure. We study the problem of existence of a universal semiregular measure on X which is invariant under a given group of bijections of X. Moreover we discuss some properties of universal, semiregular, invariant measures on groups.

This paper belongs to cylindric-algebraic model theory understood in the sense of algebraic logic. We show the existence of isomorphic but not lower base-isomorphic cylindric set algebras. These algebras are regular and locally finite. This solves a problem raised in [N 83] which was implicitly present also in [HMTAN 81]. This result implies that a theorem of Vaught for prime models of countable languages does not continue to hold for languages of any greater power.

We are interested in regressive isols, recursive functions, and the extensions of recursive functions to the isols. One of the nicest concepts that has been applied to the study of these notions is of an infinite series of isols. J. C. E. Dekker introduced infinite series of isols in [3]. With this concept one may associate with each number theoretic function u and regressive isol B a value in the isols to correspond to the series

When u is chosen as a recursive function, or as a recursive combinatorial function, many of the sums that one associates with familiar finite series may be generalized to infinite series. For example, if B is any regressive isol, then

The results presented in our paper were motivated by an interest in extending infinite series to a setting where the terms being summed may be infinite isols. In our paper, we do this in a special way, as will be described below. We would first like to briefly comment on some facts about the concept of defining an infinite series in the isols.

Le plus souvent, la logique reste une discipline à la périphérie des mathématiques, qu’elle observe de l’extérieur, sans y pénétrer vraiment. C’est un discours sur les mathematiques qui ne dit rien au mathématicien; il n’y reconnait pas son activité favorite, ni ne croit qu’elle puisse avoir une influence sur sa pratique.

L’illustration la plus extrême de cette tradition, ce sont les “reverse mathematics” de Harvey Friedman, qui connaissent le succès que l’on sait. Je veux parler ici d’une tendance toute opposée, secrétée par les développements contemporains de la théorie des modèles, qui promet des positions beaucoup plus directes.

Elle se cristalise autour de l’étude des groupes stables; l’apparition de groupes n’a rien d’inattendu dans un contexte mathématiquement signifiant: un groupe, c’est ce qui garantit une structure non-triviale (ceci n’est pas un simple argument terroriste: il y a des théoremes pour le soutenir); quant à la stabilité, c’est une hypothèse de controle structurel, au large champ d’application, et qu’on pourra dépasser quand seront résolus les problèmes posés dans le cadre stable.

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.

The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.

This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely , and using the results on Souslin forcing we show that is consistent with the existence of a Souslin tree and with the splitting number s = ℵ1. We prove that proves the additivity of measure. Also we introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + -measurability.

The theory of separably closed fields of a fixed characteristic and a fixed imperfectness degree is clearly recursively axiomatizable. Ershov [1] showed that it is complete, and therefore decidable. Later it became clear that this theory also has the prime extension property in a suitable language (cf. [4, Proposition 1]); hence it admits quantifier elimination. The purpose of this work is to give an explicit, primitive recursive procedure for such quantifier elimination in the case of a finite imperfectness degree.

In this paper the theory of the core model K is applied to study certain combinatorial principles. These principles concern the existence of families of almost disjoint functions. The first, the transversal hypothesis, is defined as follows.

Definition. The transversal hypothesis for κ, T(κ), is the following assertion:

There is a sequence 〈fν: ν < κ+〉 such that

(a) fν: κ → κ regressively for ν < κ+, and

(b) if ν < ξ < κ+, then there is γ < κ such that fν(α) ≠ fξ(α) whenever γ < α < κ.

T(κ) is a simple consequence of the Kurepa hypothesis for κ, i.e. the assertion, KH(κ), that there is a family F ⊂ P(κ) such that and card({X ∩ α: X ϵ F}) ≤ α for ω < α < κ.

The second principle to be studied, the weak Kurepa hypothesis, is a statement of strength intermediate between the Kurepa and transversal hypotheses.

Definition. The weak Kurepa hypothesis for κ, wKH(κ), is the following assertion:

There is a sequence 〈bν: ν < κ+〉 such that

(a) bν ⊂ κ for ν < κ+, and

(b) for each limit λ < κ there is Fλ: {bν ∩ λ: ν < κ+} → λ such that setting fν(λ) = Fλ(bν ∩ λ) for ν < κ+ and limit λ < κ, if ν < ξ < κ+ there is γ < κ such that fν(λ) ≠ fξ(λ) whenever γ < λ < κ and λ is a limit.

The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory.

We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2 α , then no structure theorem is expected (cf. [Sh1]).

O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set.

In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension.

In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.

Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V () of set theory. If we observe G outside V (), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V (). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.

The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

Let K be an algebraic number field and IK the ring of algebraic integers in K. *K and *IK denote enlargements of K and IK respectively. Let x Є *K – K. In this paper, we are concerned with algebraic extensions of K(x) within *K. For each x Є *K – K and each natural number d, YK(x,d) is defined to be the number of algebraic extensions of K(x) of degree d within *K. x Є *K – K is called a Hilbertian element if YK(x,d) = 0 for all d Є N, d > 1; in other words, K(x) has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural number ω, 2 ω P ω and 2 ω (ω 3 + 1) are Hilbertian elements in *Q, where p ω is the ωth prime number.

A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).

In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.

Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

§0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, “because we have proofs!” The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. The next question, naturally, is why we believe the axioms, and here the response will usually be that they are “obvious”, or “self-evident”, that to deny them is “to contradict oneself” or “to commit a crime against the intellect”. Again, the more sophisticated might prefer to say that the axioms are “laws of logic” or “implicit definitions” or “conceptual truths” or some such thing.

Unfortunately, heartwarming answers along these lines are no longer tenable (if they ever were). On the one hand, assumptions once thought to be self-evident have turned out to be debatable, like the law of the excluded middle, or outright false, like the idea that every property determines a set. Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters. In such cases, we find the methodology has more in common with the natural scientist's hypotheses formation and testing than the caricature of the mathematician writing down a few obvious truths and preceeding to draw logical consequences.

The central problem in the philosophy of natural science is when and why the sorts of facts scientists cite as evidence really are evidence. The same is true in the case of mathematics. Historically, philosophers have given considerable attention to the question of when and why various forms of logical inference are truth-preserving. The companion question of when and why the assumption of various axioms is justified has received less attention, perhaps because versions of the “self-evidence” view live on, and perhaps because of a complacent if-thenism.

An r.e. degree a ≠ 0, 0′ is called strongly noncappable if it has no inf with any incomparable r.e. degree. We show the existence of a high strongly noncappable degree.

A conservative extension of first order logic, suitable for circumscriptive inference, is introduced.

This paper is divided into three sections. §1 consists of an argument against the validity of Berry's paradox; §2 consists of supporting arguments for the thesis presented in §1; and §3 examines the possibility of re-establishing the paradox.

Berry's paradox, a semantic antinomy, is described on p. 4 of the textbook [4] as follows: