Shepherdson and Cohen independently showed that (if there is any transitive model of ZFC) there is a least transitive model of ZFC. We can ask the same question for theories extending ZFC. For some fixed set theory T, does T have a least transitive model? I will look at this question where T is a second-order set theory. Two major second-order set theories of interest are Gödel–Bernays set theory GBC and Kelley–Morse set theory KM. The weaker of the two is GBC, which is conservative over ZFC while KM is much stronger.
As an immediate corollary of the Shepherdson–Cohen result we get that there is a least transitive model of GBC. The case for KM is more difficult and indeed, has a negative answer. I will show that there is no least transitive model of KM. Along the way we will build Gödel's constructible universe above sets and into the proper classes, unroll models of second-order set theory into first-order models, and dip our toes into Barwise theory and the admissible cover. Time permitting I will mention some results and open questions about GBC + Elementary Transfinite Recursion, which is intermediate between GBC and KM in strength.