Minimal models for second-order set theories

by kamerynwilliams

This is a contributed talk at the 2018 ASL North American Annual Meeting on 17 May 2018.

Shepherdson and, independently, Cohen showed that there is a least transitive model of \mathsf{ZFC}, i.e. a transitive model of \mathsf{ZFC} which is contained inside every transitive model of \mathsf{ZFC}. An analogous question can be asked of other set theories. I will consider second-order set theories, those which have both sets and classes as their objects. It was known to Shepherdson that von Neumann–Bernays–Gödel set theory \mathsf{NBG} has a smallest transitive model. I will show that this phenomenon fails for stronger second-order set theories: there is no least transitive model of Kelley–Morse set theory \mathsf{KM}. Indeed, there is no least transitive model of \mathsf{NBG} + \Pi^1_1-Comprehension, nor any computably enumerable extension thereof. On the other hand, fragments of \mathsf{NBG} + Elementary Transfinite Recursion, which sit between \mathsf{NBG} and \Pi^1_1-Comprehension in consistency strength, do have least transitive models.