Is there a least transitive model of Kelley-Morse set theory?

This is a talk at the CUNY Set Theory Seminar on 2 October 2015.

It’s well-known that there is a least transitive model of ZFC. Namely, $L_\alpha$, where $\alpha$ is the least ordinal which is the $\mathrm{Ord}$ of a model of set theory is contained in every transitive model of ZFC. With a little bit of effort, one can extend this to see that there is a least transitive model of GBC; its first-order part is $L_\alpha$ and its second-order part is the definable classes. Can we extend this result further to get that there is a least transitive model of KM? The purpose of this talk is to answer this question in the negative.