Independence for types in algebraically closed valued fields
Deirdre Haskell · Cambridge University Press eBooks · 2007
Introduction: the historical context. My goal in this article, as it was in the lecture at the Logic Colloquium in Athens, is to survey the notion of independence of types, a fundamental tool in the area of stability theory, and see different ways in which it can be realized in a particular example of an unstable theory, the theory of algebraically closed valued fields. I thank the anonymous referee for many comments which have significantly improved this article. Of course, all remaining errors are my own. The notion of independence was first formulated by Shelah [Sh] in the 1970's in the context of classification theory. The motivating problem was the following. problem . Given a theory T and a cardinal λ > | T |, let I ( T , λ) be the number of models of T of cardinality λ, up to isomorphism. What can the function I ( T , λ) be? The first observation is that there is the following fundamental dichotomy. Suppose T is unstable; that is, for every uncountable λ there is a parameter set A with | A | ≤ λ such that the number of types over A is 2 λ . In this case, I ( T , λ) = 2 λ ; the maximum possible value. Suppose T is stable. Then there are different possibilities for I ( T , λ), so one can look for further conditions on the theory which will serve to determine the value that I ( T , λ) takes.