Large cardinals, structural reflection, and the HOD Conjecture

TL;DR

This paper introduces new large cardinal notions called exacting and ultraexacting cardinals, explores their consistency and implications for the structure of the universe of sets, and discusses their impact on the HOD Conjecture and related hypotheses.

Contribution

It defines and analyzes exacting and ultraexacting cardinals, showing their consistency relative to known large cardinals and their implications for the structure of HOD and the large cardinal hierarchy.

Findings

Ultraexacting cardinals are consistent with ZFC relative to I0 embeddings.

Existence of exacting cardinals implies V ≠ HOD, surpassing current large cardinal hierarchy.

Existence of exacting cardinals above extendible cardinals refutes Woodin's HOD and Ultimate-L conjectures.

Abstract

We introduce exacting cardinals and a strengthening of these, ultraexacting cardinals. These are natural large cardinals defined equivalently as weak forms of rank-Berkeley cardinals, strong forms of J\'onsson cardinals, or in terms of principles of structural reflection. However, they challenge commonly held intuition on strong axioms of infinity. We prove that ultraexacting cardinals are consistent with Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) relative to the existence of an I0 embedding. However, the existence of an ultraexacting cardinal below a measurable cardinal implies the consistency of ZFC with a proper class of I0 embeddings, thus challenging the linear--incremental picture of the large cardinal hierarchy. We show that the existence of an exacting cardinal implies that V is not equal to HOD (G\"odel's universe of Hereditarily Ordinal Definable sets), showing that these cardinals surpass the current hierarchy of large cardinals consistent with ZFC. Moreover, we prove that the existence of an exacting cardinal above an extendible cardinal implies the "V is far from HOD" alternative of Woodin's HOD Dichotomy. In particular, it follows that the consistency of ZFC with an exacting cardinal above an extendible cardinal would refute Woodin's HOD Conjecture and Ultimate-L Conjecture. Finally, we show that the consistency of ZF with certain large cardinals beyond choice implies the consistency of ZFC with the existence of an exacting cardinal above an extendible cardinal.