The discovery of exacting and ultraexacting cardinals emerged from collaborative research by mathematicians at the Vienna University of Technology and the University of Barcelona1. This breakthrough was detailed in a non-peer-reviewed paper titled "Large cardinals, structural reflection, and the HOD conjecture," authored by Philipp Lücke and Joan Bagaria23. Lücke, a Heisenberg fellow and Privatdozent at the University of Hamburg4, and Bagaria, an ICREA Research Professor at the University of Barcelona56, unveiled these new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal7. Their work not only introduced novel concepts in set theory but also demonstrated how these new infinities could resolve longstanding mathematical conjectures, particularly the HOD Conjecture, which has been a fundamental question in set theory for decades23.

In a non-peer-reviewed context, this research has sparked significant interest among mathematicians for its potential to bridge gaps in understanding large-cardinal hierarchies. The innovative approach to addressing the HOD Conjecture highlights the importance of interdisciplinary collaboration in advancing theoretical mathematics.

Exacting and ultraexacting cardinals possess unique properties that set them apart in the realm of set theory. These newly discovered infinities exhibit structural reflection, a characteristic that allows them to contain copies of themselves within their own structure1. This self-referential property is particularly intriguing as it suggests a form of mathematical recursion at the level of large cardinals.

Ultraexacting cardinals, in particular, demonstrate even more remarkable traits. Their existence below a measurable cardinal implies the consistency of Zermelo-Fraenkel Set Theory with Choice (ZFC) with a proper class of I0 embeddings2. This property not only expands our understanding of mathematical consistency but also provides new tools for exploring the intricate relationships between different types of large cardinals. The discovery of these cardinals opens up exciting possibilities for resolving long-standing conjectures in set theory and potentially reshaping our understanding of the foundations of mathematics3.

The discovery of exacting and ultraexacting cardinals represents a significant advancement in set theory and large cardinal theory. These new infinities have profound implications for mathematical research and our understanding of the foundations of mathematics.

• Exacting cardinals provide a new perspective on structural reflection in the large-cardinal hierarchy, offering insights into the nature of infinity below the first measurable cardinal12.

• The existence of an exacting cardinal implies that V ≠ HOD, where V is the universe of all sets and HOD is Gödel's universe of Hereditarily Ordinal Definable sets32. This result has important consequences for the HOD Conjecture, a longstanding problem in set theory.

• Ultraexacting cardinals, being even larger than exacting cardinals, have more far-reaching implications. Their existence below a measurable cardinal implies the consistency of Zermelo-Fraenkel Set Theory with Choice (ZFC) with a proper class of I0 embeddings1.

• These new cardinals bridge gaps in the large-cardinal hierarchy, potentially providing new tools for resolving other open problems in set theory and related fields4.

• The self-referential nature of exacting and ultraexacting cardinals, containing copies of themselves within their structure, introduces novel concepts in mathematical logic and set theory4.

• This discovery challenges the traditional classification of infinities, as these new cardinals don't clearly fit into existing categories of infinities that work with or without the Axiom of Choice4.

• The research, conducted by Philipp Lücke and Joan Bagaria, demonstrates the importance of collaborative efforts in advancing theoretical mathematics567.

• While the findings are currently presented in a non-peer-reviewed paper, they have already generated significant interest in the mathematical community, potentially leading to new research directions and collaborations2.

The discovery of exacting and ultraexacting cardinals has significant implications for both infinity concepts and mathematics as a whole. These new types of infinity challenge the linear-incremental picture of the large cardinal hierarchy, suggesting a more complex structure to the mathematical universe1. Their existence implies that V ≠ HOD (where V is the universe of all sets and HOD is Gödel's universe of Hereditarily Ordinal Definable sets), potentially disproving the Weak HOD Conjecture and the Weak Ultimate-L Conjecture2.

This breakthrough also impacts our understanding of structural reflection in mathematics, providing new tools for exploring set theory and its foundations3. The consistency of ultraexacting cardinals with Zermelo-Fraenkel Set Theory and the Axiom of Choice (ZFC) relative to the existence of an I0 embedding opens up new avenues for research in mathematical logic and set theory2. These discoveries may lead to novel approaches in solving long-standing mathematical problems and could influence related fields such as theoretical physics and computer science, where concepts of infinity play crucial roles45.