Invariant Version of Cardinality Quantifiers in Superstable Theories

Abstract

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula.