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 $\kappa \le \lambda$, then $$\underset{|A|=\lambda }{\sup }|S^{\kappa }(A)|=(\underset{|A|=\lambda }{\sup }|S^1(A)|{)}^{\kappa }$$ 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.

Cited by in corpus (2)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Symmetry in abstract elementary classes with amalgamation