March 22
Joel David Hamkins, Oxford University
Kelley-Morse set theory does not prove the class Fodor Principle

I shall discuss recent joint work with Victoria Gitman and Asaf Karagila, in which we proved that Kelley-Morse set theory (which includes the global choice principle) 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, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq{\rm Ord}$ that there is a class function $F:{\rm Ord}\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a sequence of classes $A_n$, each containing a class club, but the intersection of all $A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.