The Axiom of Choice in the $\kappa$-Mantle

Abstract

Usuba has asked whether the $\kappa$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}\kappa$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $\kappa$ is a Mahlo cardinal, $\kappa=\omega_1$ and where $\kappa$ is the successor of a regular uncountable cardinal.