Will Boney: Computing the Number of Types of Infinite Length, Part 1

Model Theory Seminar
Will Boney
Carnegie Mellon University

Title:  Computing the Number of Types of Infinite Length, Part 1.

Abstract:  Roughly, the type of an element over a given domain is the best description of the element possible using formulas from the language and parameters from the domain.  Counting the largest number of types of finite tuples possible over a domain of fixed size give important information about the theory and characterizes its place in relation to certain dividing lines, such as stability.  It is known that it is enough to check stability just for 1-types.  We generalize this result to types of infinite tuples of elements by calculating supremum number of types of infinite tuples over a domain of fixed size from the number of 1-types.  In particular, for $\kappa \leq \alpha$, we show

$$\sup_{|A| = \lambda} |S^\kappa(A)| = \left( \sup_{|A| = \lambda} |S^1(A)| \right)^\kappa$$
Building on this, we consider nonalgebraic types.  Unfortunately, no computation like this is possible for these types.  Fortunately, there is a natural generalization of these types that we call \emph{strongly separative} that does admit an upper bound.  The proofs are carried out in the context of Abstract Elementary Classes and Galois types, which allow the results to hold for nonelementary classes as well.  We aim to make this talk accessible to all and will provide all relevant definitions; five weeks of model theory should be sufficient.
Date: Monday, September 30, 2013
Time: 5:00 pm
Location: Wean 8220

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