Superstructures on Set Theory

We will discuss a number of ways to put a "superstructure" over the usual set theory ZFC. These will include von Neumann-G\"odel-Bernays (predicative) class theory, Kelley-Morse (impredicative) class theory, the set theory of Ackermann with sets and classes and powerful extensions of Jensen's NFU (New Foundations with urelements) which may not seem at first glance to be superstructures on ZFC. Positive set theory may also be discussed, if time permits. The idea of viewing set theories with universal set such as NFU or positive set theory as alternative superstructures to the usual theory of classes has been discussed earlier by Roland Hinnion. It has been a motivating concept behind our development of a sequence of strong extensions of NFU.