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 $\omega \le \lambda \le \text{Ord}$ 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.

18 pages. Commentary about this article can be made at http://jdh.hamkins.org/km-does-not-prove-class-fodor

References in corpus (3)

• The exact strength of the class forcing theorem
• The Structure of Models of Second-order Set Theories
• Force to change large cardinal strength

Cited by in corpus (3)

• Cichoń's Diagram and Localisation Cardinals
• Set-Theoretic Blockchains
• Broad Infinity and Generation Principles