The Axiom of Choice in the $κ$-Mantle
Andreas Lietz · arXiv (Cornell University) · 2024
Usuba has asked whether the $κ$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}κ$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $κ$ is a Mahlo cardinal, $κ=ω_1$ and where $κ$ is the successor of a regular uncountable cardinal.