Kelley-Morse set theory does not prove the class Fodor principle

Abstract

We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\text{Ord}$ that there is a class function $F:\text{Ord}\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteq\omega\times\text{Ord}$, such that each section $A_n=\{\alpha\mid (n,\alpha)\in A\}$ contains a class club, but $\bigcap_n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.