A groundbreaking paper titled Large Cardinals, Structural Reflection, and the HOD Conjecture has introduced the concepts of "precise cardinals" and their enhanced counterpart, "hyperprecise cardinals."

These novel large cardinal concepts challenge established intuitions about the linear hierarchy of large cardinals and the HOD Conjecture, offering new perspectives for research in infinite axioms.

The study defines precise cardinals and hyperprecise cardinals through equivalent frameworks, including weak forms of rank Berkeley cardinals, strong forms of Jónsson cardinals, and structural reflection principles. Despite their natural formulation, these cardinals challenge conventional expectations about strong axioms of infinity.

The paper demonstrates that hyperprecise cardinals are consistent with Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), assuming the existence of l₀ embeddings. However, if a hyperprecise cardinal exists below a measurable cardinal, it implies the consistency of ZFC with proper class l₀ embeddings, defying the traditional linear increment paradigm of large cardinal hierarchies.

The existence of precise cardinals implies V ≠ HOD, where V represents the universe of all sets and HOD refers to Gödel’s constructible universe defined by hereditary ordinal definability. This suggests that these cardinals transcend the conventional large cardinal hierarchy within ZFC consistency. Furthermore, the paper argues that precise cardinals above extendible cardinals can refute Woodin's HOD Conjecture and the Ultimate L Conjecture, expanding the boundaries of large cardinal research.

The Scott Inconsistency Theorem states that ZFC combined with V=L (the constructible universe axiom) is inconsistent with the existence of non-trivial elementary embeddings from V to a transitive class M. Such embeddings derive from the existence of measurable cardinals. Keisler further refined this by showing that the critical point of such embeddings must be measurable. Ulam originally defined measurable cardinals as those admitting a complete, κ-complete, two-valued measure.

The Scott-Keisler theorem highlights that some large cardinals fundamentally conflict with V=L. The new concepts of precise and hyperprecise cardinals are developed to align with ZFC under l₀ embeddings, serving as counterparts to weak large cardinals compatible with V=L. By restricting the domain of elementary embeddings, these cardinals emerge as a natural evolution from incompatible counterparts like rank Berkeley cardinals.

The existence of precise cardinals introduces a novel division within the large cardinal hierarchy, distinct from ZFC-consistent hierarchies. This challenges the intuition that all strong axioms of infinity should be compatible with V=HOD. Moreover, the extreme interactions between these cardinals and other large cardinal axioms further disrupt the traditional linear hierarchy.

Woodin's HOD Conjecture posits that the universe V is "close" to HOD, claiming that ZFC combined with the existence of an extendible cardinal implies that sufficiently large regular cardinals are not ω-strongly measurable in HOD. This study demonstrates that precise cardinals combined with extendible cardinal axioms can directly contradict this conjecture.

The discovery of hyperprecise cardinals redefines their interaction with the traditional linear structure of large cardinals. These cardinals amplify the effects of other large cardinals, extending Woodin’s Icarus hierarchy and prompting a reevaluation of large cardinal hierarchies. By expanding the boundaries of current frameworks, these findings offer profound insights into the nature of infinity and its mathematical representation.