Rather classless models of set theory and second-order set theory

This is a talk at the CUNY Set Theory Seminar on 13 March 2015. Notes can be seen at my old site.

A model M of ZFC is rather classless if every class of M all of whose bounded initial segments are in M is definable in M. In this talk, we will construct rather classless end extensions for every countable model of set theory. As an application of this construction, we will see that there are models of ZFC with precisely one extension to a model of GBC and that there are models of set theory which admit no extension to a model of GBC. If time permits, we will look at some related constructions with models of KM + the axiom schema of class choice.