Simple Theories

This paper continues work in [ 111. In [ 141, Shelah studies a class of first order theories which he calls simple, his point being to find meaningful dividing lines within the class of unstable theories. Simple theories are defined to be complete theories without the tree property (see below). They form a subclass of theories without the strict order property and include stable theories. The prototypical example of a simple unstable theory is the theory of the random graph. In [14], it was shown that in simple theories, forking (as defined in [ 131) behaves reasonably well as a notion of dependence. Striking progress was made in [l l] where it was shown that forking satisfies all the properties of forking in stable theories, except the boundedness of the set of nonforking extensions of a type. One of the main aims of this paper is to find the right analogue for simple theories of the stationarity of types over models (or more generally strong types) in stable theories. We succeed in doing this by proving the Independence Theorem over a model for simple theories. This states that if A4 is a model, A >M, B >M, A is independent from B over A4, p E S(M), and p1 E S(A), p2 E S(B) are both nonforking extensions of p, then there is q E S(A U B) which extends both pl and p2 and is also a nonforking extension of p. In fact we also show conversely that any theory equipped with a notion of independence satisfying all the basic algebraic axioms, together with the Independence Theorem over a model, must be simple, and that moreover the notion of independence must coincide with nonforking. We consider this characterisation of simple theories as showing the simple/nonsimple dividing line to be meaningful. The “correct” notion of strong type in simple theories appears to be a notion originating