Fodor's lemma can fail everywhere

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, Fodor's lemma can fail universally and the club filter on regular cardinals can lack even basic completeness, with these failures being precisely controllable.

Contribution

It establishes the equiconsistency of the universal failure of Fodor's lemma with ZF and shows that the club filter can be non-$\sigma$-complete everywhere, with precise control over these failures.

Findings

Fodor's lemma can fail everywhere under certain assumptions.

The club filter on every regular cardinal can lack $\sigma$-completeness.

These failures are shown to be controllable in a very precise manner.

Abstract

We show that it is equiconsistent with $\mathsf{ZF}$ that Fodor's lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even $\sigma$-complete. Moreover, these failures can be controlled in a very precise manner.