Title: Least models of second-order set theories,
Abstract: The main theorems of this paper are (1) there is no least transitive model of
Kelley--Morse set theory $\mathsf{KM}$ and (2) there is a least
$\beta$-model---that is, a transitive model which is correct about which of its
classes are well-founded---of Gödel--Bernays set theory $\mathsf{GBC}$ +
Elementary Transfinite Recursion. Along the way I characterize when a countable
model of $\mathsf{ZFC}$ has a least $\mathsf{GBC}$-realization and show that no
countable model of $\mathsf{ZFC}$ has a least $\mathsf{KM}$-realization. I also
show that fragments of Elementary Transfinite Recursion have least
$\beta$-models and, for sufficiently weak fragments, least transitive models.
These fragments can be separated from each other and from the full principle of
Elementary Transfinite Recursion by consistency strength. The main question
left unanswered by this article is whether there is a least transitive model of
$\mathsf{GBC}$ + Elementary Transfinite Recursion.