We develop the theory of domination by stable types and stable weight in an arbitrary theory.

§1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular (...) algebraic structures which have been studied recently turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T|. (shrink)

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

The new result of this paper is that for θ-stable we have S1[θ] = D[θ, L, ∞]. S1 is Hrushovski's rank. This is an improvement of a result of Kim and Pillay, who for simple theories under the assumption that either of the ranks be finite obtained the same identity. Only the first equality is new, the second equality is a result of Shelah from the seventies. We derive it by studying localizations of several rank functions, we get the followingMain (...) Theorem. Suppose that μ is regular satisfying μ ≥ |T|+, p is a finite type, and Δ is a set of formulas closed under Boolean operations. If either R[p, Δ, μ+] < ∞ or p is Δ-stable and μ satisfies “for every sequence {μi : i < |Δ| + ℵ0} of cardinals μi < μ we have that equation image holds”, then S[p, Δ, μ+] = D[p, Δ, μ+] = R[p, Δ, μ+].The S rank above is a localized version of Hrushovski's S1 rank. This rank, as well as our systematic use of local stability, allows us to get a more conceptual proof of the equality of D and R, which is an old result of Shelah. A particular case of the theorem offers a new sufficient condition for the equality of S1 and D[·, L, ∞]. We also manage, due to a more general approach, to avoid some combinatorial difficulties present in Shelah's original exposition. (shrink)

The notion of Mathematics as Ontology (as defined by Badiou in his work) is brought into question from a working mathematician's perspective. Notions of independence in set theory and model theory are contrasted with the original equation Mathematics=Ontology. The author builds an extension of mathematical ontology from set theory to a foliated, étale, setting.

We introduce a new device in the study of abstract elementary classes : Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. We show:Theorem 0.1 An AEC K is fully \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa = \beth _{\kappa } > \text {LS}$$\end{document}. If K is Galois stable, then the (...) class of κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Galois saturated models of K admits an independence notion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory. (shrink)

We define a formula φ in a first-order language L , to be an equation in a category of L -structures K if for any H in K , and set p = {φ;i ϵI, a i ϵ H} there is a finite set I 0 ⊂ I such that for any f : H → F in K , ▪. We say that an elementary first-order theory T which has the amalgamation property over substructures is equational if every quantifier-free (...) formula is equivalent in T to a boolean combination of equations in Mod, the category of models of T with embeddings for morphisms. Thus, we develop a theory of independence with respect to equations in general categories of structures, which is similar to the one introduced in stability but which, in our context, has an algebraic character. (shrink)

Shelah's theory of forking is generalized in a way which deals with measures instead of complete types. This allows us to extend the method of forking from the class of stable theories to the larger class of theories which do not have the independence property. When restricted to the special case of stable theories, this paper reduces to a reformulation of the classical approach. However, it goes beyond the classical approach in the case of unstable theories. Methods from ordinary forking (...) theory and the Loeb measure construction from nonstandard analysis are used. (shrink)

We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.