Thesis Work: “persistence and Regularity in Unstable Model Theory”

Historically one of the great successes of model theory has been Shelah’s stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity classes according to a priori set-theoretic criteria (e.g. counting types over sets) in fact pushes down to reveal a very rich and entirely model-theoretic structure theory for the classes involved: what we now call stability, superstability, and ω-stability, as well as the dichotomy between independence and strict order in unstable theories. The success of the program may be measured by the fact that the original set-theoretic criteria are now largely passed over in favor of definitions which mention ranks or combinatorial properties of a particular formula. Because of this shift, Keisler’s 1967 order (defined below) may strike the modern reader as an anachronism. It too seeks to coarsely classify first-order theories in terms of a more settheoretic criterion, the difficulty of producing saturated regular ultrapowers, but its structure has remained largely open. Partial results from the 70s suggest a mine of perhaps comparable richness, one which has remained largely inaccessible to current tools. Keisler’s criterion of choice, saturation of regular ultrapowers, is natural for two reasons. First, when the ultrapower is regular, the degree of its saturation depends only on the theory and not on the saturation of the index models. Second, ultrapowers are a natural context for studying compactness, and Keisler’s order can be thought of as studying the fine structure of compactness by asking: what families of consistent types are realized or omitted together in regular ultrapowers? Thus the relative difficulty of realizing the types of T1 versus those of some T2 in regular ultrapowers gives a measure of the combinatorial complexity of the types each Ti is able to describe.