Kelley–Morse set theory does not prove the class Fodor principle
Tom 254 / 2021
Streszczenie
We show that Kelley–Morse KM set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to {\rm Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, for every $\omega \leq \lambda \leq {\rm Ord}$, it is relatively consistent with KM that there is a class function $F:{\rm Ord} \to \lambda $ that is not constant on any stationary class, and moreover $\lambda $ is the least ordinal for which such a counterexample function exists. As a corollary of this result, it is consistent with KM that there is a class $A\subseteq \omega \times {\rm Ord}$ 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.
