Cofinality spectrum theorems in model theory, set theory and general topology

by Malliaris and Shelah. [MiSh:998]
J American Math Soc, 2016
We solve a set-theoretic problem (p=t) and show it has consequences for the classification of unstable theories. That is, we consider what pairs of cardinals (kappa_1, kappa_2) may appear as the cofinalities of a cut in a regular ultrapower of linear order, under the assumption that all symmetric pre-cuts of cofinality no more than the size of the index set are realized. We prove that the only possibility is (kappa, kappa^+) where kappa is regular and kappa^+ is the cardinality of the index set I . This shows that unless |I| is the successor of a regular cardinal, any such ultrafilter must be |I|^+-good. We then connect this work to the problem of determining the boundary of the maximum class in Keisler's order. Currently, SOP_3 is known to imply maximality. Here, we show that the property of realizing all symmetric pre-cuts characterizes the existence of paths through trees and thus realization of types with SOP_2 (it was known that realizing {all} pre-cuts characterizes realization of types with SOP_3). Thus whenever lambda is not the successor of a regular cardinal, SOP_2 is lambda-maximal in Keisler's order. Moreover, the question of the full maximality of SOP_2 is reduced to either constructing a regular ultrafilter admitting the single asymmetric cut described, or showing one cannot exist.

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