### The structure of models of second-order set theories

This is my PhD dissertation.

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Abstract.This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the 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 model ofZFC, where T is a reasonable second-order set theory such asGBCorKM, showing that it has 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 of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski fromKMto weaker theories. They showed that every model ofKMplus the Class Collection schema “unrolls” to a model ofZFC^{−}with a largest cardinal. Icalculate the theories of the unrolling for a variety of second-order set theories, going as weak asGBC + 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 fromGBCtoKM. This hierarchy is orderedfirst by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such asKMor-CA—do not have least transitive models, while weaker theories—fromGBCtoGBC + ETR_{Ord}—do have least transitive models.

`@PHDTHESIS{Williams:dissertation,`

author = {Kameryn J Williams},

title = {The structure of models of second-order set theories},

school = {The Graduate Center, The City University of New York},

year = {2018},

eprint = {1804.09526},

archivePrefix ={arXiv},

primaryClass = {math.LO},

url = {https://kamerynblog.wordpress.com/2018/04/25/the-structure-of-models-of-second-order-set-theories},

}