Strong second-order set theories do not have least transitive models

by kamerynwilliams

This will be a talk at the University of Pennsylvania Logic and Computation Seminar on Monday, November 13. See also their page for the talk. Slides available here.

Shepherdson and Cohen independently showed that (if there is any transitive model of \mathsf{ZFC}) there is a least transitive model of \mathsf{ZFC}. That is, there is a transitive set M so that (M,\in) \models \mathsf{ZFC} and if N is any transitive model of \mathsf{ZFC} then M \subseteq N. We can ask the same question for theories extending \mathsf{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 \mathsf{GBC} and Kelley–Morse set theory \mathsf{KM}. The weaker of the two is \mathsf{GBC}, which is conservative over \mathsf{ZFC} while \mathsf{KM} is much stronger.

As an immediate corollary of the Shepherdson–Cohen result we get that there is a least transitive model of \mathsf{GBC}. The case for \mathsf{KM} is more difficult and indeed, has a negative answer. I will show that there is no least transitive model of \mathsf{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 \mathsf{GBC} + Elementary Transfinite Recursion, which is intermediate between \mathsf{GBC} and \mathsf{KM} in strength.