Kameryn J. Williams successfully defended his dissertation under my supervision at the CUNY Graduate Center on April 6th, 2018, earning his Ph.D. degree in May 2018. He has accepted a position in mathematics at the University of Hawaii, to begin Fall 2018.

What a pleasure it was to work with Kameryn, an extremely talented mathematician with wide interests and huge promise.

Kameryn J Williams | MathOverflow | ar$\chi$iv

Kameryn J. Williams, The Structure of Models of Second-order Set Theories, Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2018. arXiv:1804.09526.

Abstract.This dissertation is a contribution to the project of second-order set theory, which has seena revival in recent years. The approach is to understand second-order set theory by studyingthe structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable modelof ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that ithas a rich structure. In particular, every countable partial order embeds into this structure.Moreover, we can arrange so that these embedding preserve the existence/nonexistence ofupper bounds, at least for finite partial orders. Second I generalize some constructions ofMarek and Mostowski from KM to weaker theories. They showed that every model of KMplus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. Icalculate the theories of the unrolling for a variety of second-order set theories, going asweak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broadselection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is orderedfirst by the complexity of the properties allowed in the recursions and second by the allowedheights of the recurions. Fourth, I investigate the question of which second-order set theorieshave least models. I show that strong theories—such as KM or $\Pi^1_1$-CA—do not have leasttransitive models, while weaker theories—from GBC to GBC + ETR${}_{\text{Ord}}$—do have least transitivemodels.

In addition to his dissertation work and the research currently arising out of it, Kameryn has undertaken a number of collaborations with various international research efforts, including the following:

- He is a co-author on The exact strength of the class forcing theorem.
- He is co-author on a current joint project with Miha Habič, myself, Daniel Klausner and Jonathan Verner concerning the nonamalgamation phenomenon in the generic multiverse of a countable model of set theory.
- He is co-author on a current joint project with myself and Philip Welch concerning the universal $\Sigma_1$-definable finite sequence, an analogue of the universal finite set, but for the constructible universe.