Computing the Number of Types of Infinite Length

Abstract

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if $\kappa \leq \lambda$, then $$\sup_{|A| = \lambda} |S^\kappa(A)| = (\sup_{|A| = \lambda} |S^1(A)|)^\kappa$$ We show that this holds for any abstract elementary class with $\lambda$ amalgamation, but it is new for first order theories when $\kappa$ is infinite. No such calculation is possible for nonalgebraic types. We introduce a generalization of nonalgebraic types for which the same upper bound holds.