Large cardinals beyond HOD

Abstract: Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the set-theoretic universe V is not equal to G\"odel's subuniverse of Hereditarily Ordinal Definable (HOD) sets. We prove that the existence of an ultraexacting cardinal is equiconsistent with the well-known axiom I0; moreover, the existence of ultraexacting cardinals together with other standard large cardinals is equiconsistent with generalizations of I0 for fine-structural models of set theory extending $L(V_{\lambda+1})$. We prove tight bounds on the strength of exacting cardinals, placing them strictly between the axioms I3 and I2. The argument extends to show that I2 implies the consistency of Vop\v{e}nka's Principle together with an exacting cardinal and the HOD Hypothesis. In particular, we obtain the following result: the existence of an extendible cardinal above an exacting cardinal does not refute the HOD Hypothesis. We also give several new characterizations of exacting and ultraexacting cardinals; first in terms of strengthenings of the axioms I3 and I1 with the addition of Ordinal Definable predicates, and finally also in terms of principles of Structural Reflection which characterize exacting and ultraexacting cardinals as natural two-cardinal forms of strong unfoldability.