These lecture notes originated in a seminar on Model Theory that I gave in the academic years 2005–06 and 2006–07 at the Department of Logic, History and Philosophy of Science of the University of Barcelona. I had presented some previous work on the basic notions of simple theories in July 2002 in the Simpleton Workshop held at the Centre International de Rencontres Mathématiques, Luminy (Marseille), which was subsequently published as [8]. A more extended version, including the exposition of stable theories, was the topic of a tutorial entitled Advanced Stability Theory that I taught in the Modnet Summer School that took place at the University of Freiburg in April 2006. And in preparing the material, I also drew on some courses on these topics given at the Universidad de los Andes, Bogotá, in August 2000 and in August 2004.

The notes are based on the work of many model theorists. The names of John T. Baldwin, Ehud Hrushovski, Byunghan Kim, Daniel Lascar, Ludomir Newelski, Anand Pillay, Bruno Poizat, Saharon Shelah, Frank O. Wagner, and Martin Ziegler deserve special mention. I learned stability theory from Martin Ziegler and I have made as much use as I could of his short and elegant proofs, presented in his courses and in his unpublished lecture notes.

This book is not as ambitious as Frank O. Wagner's book on simple theories [41], but its pace might be more comfortable for the beginner.

Definition 7.1. Let M ⊆ A and p(x) ∈ S(A). We say that p is an heir of p ↾ M or that p inherits from M if for every φ(x, y) ∈ L(M) if φ(x, a) ∈ p for some tuple a ∈ A, then φ(x, m) ∈ p for some tuple m ∈ M. We say that p is a coheir of p ↾ M or that p coinherits from M if p is finitely satisfiable in M. The same definitions apply to global types, i.e., to the case A = ℭ. These definitions also make sense for types in infinitely many variables.

Remark 7.2. tp(a/Mb) inherits from M if and only if tp(b/Ma) coinherits from M.

Proof. It is just a matter of writing down the definitions.

Lemma 7.3.

1. 1. If p(x) ∈ S(M), then p inherits and coinherits from M.

2. 2. If M ⊆ A and p(x) ∈ S(A) coinherits from M, then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q coinherits from M.

3. 3. If M ⊆ A and p(x) ∈ S(A) inherits from M, then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q inherits from M.

Definition 11.1. The multiplicity of a type p(x) ∈ S(A) is the number Mlt(p) of its global nonforking extensions p(x) ∈ S(ℭ). If there is a proper class of global nonforking extensions of p, we say that p has unbounded multiplicity and we write Mlt(p) = ∞; otherwise we say that p has bounded multiplicity. A stationary type is a type of multiplicity 1. Thus over any B ⊇ A a stationary type p(x) ∈ S(A) has a unique nonforking extension q(x) ∈ S(B). We use the notation p|B for q.

Lemma 11.2. Let T be simple. If p ∈ S(A) is stationary, then its global nonforking extension is Definable over A.

Proof. Let p be the global nonforking extension of p, and let φ(x, y) ∈ L. We will show that p ↾ φ is A-definable. Let Δφ(y) and Δ¬φ(y) be types over A given by Corollary 5.23 for p and φ and for p and ¬φ respectively. By compactness, the conjunction ψ(y) of a finite subset of Δφ(y) is inconsistent with Δ¬φ(y). It is clear that ψ(y) defines p ↾ φ.

Corollary 11.3. Let T be simple. If types over models are stationary, then T is stable.

Proof. Lemma 11.2 implies that in this situation every global type is Definable.

Definition 16.1. Let A be a class of hyperimaginaries and let I be a set linearly ordered by <. The sequence of hyperimaginaries (ei : i ∈ I) is indiscernible over A or it is A-indiscernible if for every n < ω, for every two increasing sequences of indices i0 < … < in and j0, < … < jn,. If A is a set, in practice we may always assume that A is a single hyperimaginary. Note that all the hyperimaginaries ei are in fact of the same sort and hence we can write ei = [ai]E for a single E.

Lemma 16.2. Let d be a hyperimaginary.

1. 1. Let I, J be linearly ordered infinite sets. If (ei : i ∈ I) is a d-indiscernible sequence of hyperimaginaries, then there is a d-indiscernible sequence (ci : j ∈ J) such that for every n < ω, for every two increasing sequences of indices i0 < … < in ∈ I and j0 < … < jn ∈ J,.

2. 2. If (ei : i ∈ I) and (di : i ∈ I) are d-indiscernible sequences of hyperimaginaries and (ei : i ∈ I0) ≡d (di : i ∈ I0) for each finite subset I0 ⊆ I, then f((ei : i ∈ I)) = (di : i ∈ I) for some f ∈ Aut(ℭ/d).