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

TL;DR

The paper demonstrates that Kelley-Morse set theory cannot prove the class Fodor principle and constructs models showing its independence, revealing limitations of KM in class function regularity.

Contribution

It proves the independence of the class Fodor principle from Kelley-Morse set theory and constructs models illustrating its failure.

Findings

KM does not prove the class Fodor principle.

Models exist where regressive class functions are not constant on stationary subclasses.

The class club filter may not be $\sigma$-closed in KM models.

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.