The effects of forcing on exacting and ultraexacting cardinals

Abstract

Motivated by recent work of Aguilera--Bagaria--Lücke and Aguilera--Bagaria--Goldberg--Lücke (ABGL), we analyze various effects of set-theoretic forcing upon the classes of exacting and ultraexacting cardinals. Using Magidor support products of Prikry forcings, we prove that an I2-embedding yields a transitive, set-sized model of ZFC with a proper class of exacting cardinals. We show that under Woodin's HOD Conjecture, no exacting cardinal can be a limit of exacting cardinals. In contrast, we prove that models of ZF containing large cardinals beyond choice have forcing extensions that are models of ZFC in which a regular cardinal is a stationary limit of ultraexacting cardinals. We also show that, assuming the HOD Conjecture, an analogue of the classical Levy--Solovay theorem holds for exacting and ultraexacting cardinals. Finally, we analyze the large cardinal properties possessed by exacting cardinals in HOD. We prove that if $λ$ is exacting and $V_λ$ satisfies the HOD Hypothesis, then $λ$ has strong large cardinal properties in HOD. Carrying forward forcing constructions from ABGL, we start with a model of ZFC with an I2-embedding and produce a model of ZF in which the successor of an exacting cardinal $λ$ is extendible in all models of the form $\mathrm{HOD}_x$ for $x\subseteq λ$.