Choice principles in local mantles

TL;DR

This paper investigates the relationship between certain choice principles in the $oldsymbol{ ext{kappa}}$-mantle and large cardinal properties like inaccessibility and weak compactness, within the framework of ZFC and forcing.

Contribution

It establishes that specific choice principles in the $oldsymbol{ ext{kappa}}$-mantle follow from $oldsymbol{ ext{kappa}}$ being an inaccessible or weakly compact cardinal.

Findings

Partial choice principles in the $oldsymbol{ ext{kappa}}$-mantle are consequences of $oldsymbol{ ext{kappa}}$'s large cardinal properties.

Certain large cardinal assumptions imply the validity of specific choice principles in the $oldsymbol{ ext{kappa}}$-mantle.

The paper connects forcing, large cardinals, and the structure of the $oldsymbol{ ext{kappa}}$-mantle.

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.