Choice principles in local mantles

Abstract

Assume ZFC. Let $\kappa$ be a cardinal. A $<\kappa$-ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality $<\kappa$. The $\kappa$-mantle is the intersection of all $<\kappa$-grounds. We prove that certain partial choice principles in the $\kappa$-mantle are the consequence of $\kappa$ being inaccessible/weakly compact, and some other related facts.

19 pages. This version: Corrected some false statements (which were not used anywhere; see Footnotes 3, 7, 11, 13), in particular a statement falsely attributed to Usuba (see Footnote 3). Removed what was previously section 4, for inclusion in a separate paper. Improved exposition and made other minor edits