Abstract

We study reflection principles in Kelley-Morse set theory with urelements (KMU). We first show that First-Order Reflection Principle is not provable in KMU with Global Choice. We then show that KMU + Limitation of Size + Second-Order Reflection Principle is mutually interpretable with KM + Second-Order Reflection Principle. Furthermore, these two theories are also shown to be bi-interpretable with parameters. Finally, assuming the existence of a κ+-supercompact cardinal κ in KMU, we construct a model of KMU + Second-Order Reflection Principle where Limitation of Size fails.