After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove that second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $\kappa $ is supercompact if and only if every $\Pi ^1_1$ sentence true in a structure M (of any size) containing $\kappa $ in a language of size less than $\kappa $ is also true in a substructure $m\prec M$ of size less than $\kappa $ with $m\cap \kappa \in \kappa $.

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR.