Set Theory Seminar

April 6
Kameryn Williams, CUNY
Dissertation defense: The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The main results concern the structure of possible collections of classes which may be added to a fixed countable model of ZFC to give a model of second-order set theory. I show that this is a rich structure, with every countable partial order embedding into it. I show that given a countable model of ZFC there is never a smallest collection of classes to put on it to get a model of Kelley–Morse set theory. This implies that there is not a least transitive model of KM, in contrast to the well-known Shepherdson–Cohen theorem that there is a least transitive model of ZFC. More generally, no second-order set theory which proves the existence of the least admissible above V can have a least transitive model. On the other hand, ETR—Gödel–Bernays set theory plus the principle of Elementary Transfinite Recursion—does have a least transitive model. As an important tool towards these results I generalize a construction of Marek and Mostowski which shows that every model of KM (plus the Class Collection schema) 'unrolls to a model of a first-order set theory. I calculate the theories of the unrollings for a variety of second-order set theories, going as weak as ETR. This is used to show that being T-realizable, for a broad class of second-order set theories T, goes down to inner models.