Lecture Notes on Forking

1.1. Types. Most of the time we fix a complete countable first-order theory T in a language L, and we let M be a very saturated model of it, a “monster” model. All our sets and models are subsets and elementary submodels of M. A set is called small if it is cardinality is smaller than the saturation of M. For a set A, we let Def (A) be the boolean algebra of A-definable subsets of M, i.e. sets of the form {x ∈M : M |= φ (x, a)} for φ (x, y) ∈ L and a a tuple of elements from A. The space of types in the variable x over A, Sx (A), is the Stone dual of Def (A). It is the space of ultrafilters on Def (A), a compact Hausdorff totally-disconnected space with the clopen basis given by 〈φ (x, a)〉 = {p ∈ S (A) : φ (x, a) ∈ p}. Elements of S (M) are called global types. We will write a ≡C b if tp (a/C) = tp (b/C). Then a ≡C b if and only if there is an automorphism σ of M fixing C and such that σ (a) = b.