We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Criteres de compacite" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.

We study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (NSOP) in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, Talagrand's stability, and explain the relationship between this property and the NIP in continuous logic. Using a result of Bourgain, Fremlin, and Talagrand, we prove almost definability and Baire 1 definability of coheirs assuming the NIP. We show that a formula has (...) the strict order property if and only if there is a convergent sequence of continuous functions on the space of φ‐types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein‐Šmulian theorem. (shrink)

We define the notion has the NIP (not the independence property) in A, where A is a subset of a model, and give some equivalences by translating results from function theory. We also discuss the number of coheirs when A is not necessarily countable, and revisit the notion “ has the NOP (not the order property) in a model M”.

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the [Formula: see text]-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of [Formula: see text]-invariant types to that of [Formula: see text]-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small (...) model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs. (shrink)

We give a new characterization of _SOP_ (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has _OP_ (the order property) if and only if it has _IP_ (the independence property) or _SOP_, in several ways by characterizing various notions in functional analytic style. We point out some connections (...) between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories. (shrink)