Large Cardinals Challenge Established Views on Set Theory and the HOD Conjecture

Core Concepts

This research paper introduces exacting and ultraexacting cardinals, novel large cardinal axioms in set theory, demonstrating their consistency with the ZFC axioms and their profound implications for the structure of the set-theoretic universe, particularly challenging the HOD Conjecture.

Abstract

• Bibliographic Information: Aguilera, J. P., Bagaria, J., & Lücke, P. (2024). Large cardinals, structural reflection, and the HOD Conjecture. arXiv preprint arXiv:2411.11568.

• Research Objective: This paper introduces and investigates the properties of two new large cardinal axioms: exacting cardinals and ultraexacting cardinals. The authors aim to situate these cardinals within the existing hierarchy of large cardinals and explore their implications for the structure of the set-theoretic universe, particularly concerning the HOD Conjecture.

• Methodology: The authors employ techniques from axiomatic set theory, particularly focusing on the construction and analysis of elementary embeddings between models of set theory. They leverage established results about large cardinals, such as rank-Berkeley cardinals and I0 embeddings, to establish the consistency and properties of exacting and ultraexacting cardinals.

• Key Findings:

  • Exacting and ultraexacting cardinals are consistent with the ZFC axioms, relative to the existence of an I0 embedding.
  • Exacting cardinals are incompatible with the universe of sets being equal to the class of hereditarily ordinal definable sets (HOD), a property not shared by previously known large cardinals consistent with ZFC.
  • Ultraexacting cardinals, while not stronger than I0 in consistency strength, exhibit unusual interactions with other large cardinals, significantly amplifying their consistency strength.
  • The existence of an exacting cardinal above an extendible cardinal refutes Woodin's HOD Conjecture and the Weak Ultimate-L Conjecture.
  • The authors establish the consistency of ZFC with an exacting cardinal above an extendible cardinal, assuming the consistency of ZF with certain large cardinals beyond the Axiom of Choice.

• Main Conclusions: Exacting and ultraexacting cardinals represent a significant discovery in set theory, challenging established views about the nature of large cardinals and their relationship with the HOD Conjecture. Their existence suggests a more complex and nuanced picture of the large cardinal hierarchy than previously understood.

• Significance: This research has profound implications for the foundations of mathematics, particularly for our understanding of the structure of the set-theoretic universe and the limits of definability. The results challenge long-held assumptions about the nature of large cardinals and their compatibility with principles like V=HOD.

• Limitations and Future Research: The authors acknowledge that the consistency results rely on the assumption of strong large cardinal axioms, whose own consistency remains an open question. Further research could explore the potential for weakening these assumptions or investigating the properties of exacting and ultraexacting cardinals in alternative set-theoretic frameworks.

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Large cardinals, structural reflection, and the HOD Conjecture

by Juan... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11568.pdf

Deeper Inquiries

How do exacting and ultraexacting cardinals relate to other large cardinal notions that are incompatible with the Axiom of Choice, such as Reinhardt cardinals?

Exacting and ultraexacting cardinals, while compatible with the Axiom of Choice, are indeed closely related to larger cardinal notions incompatible with it, particularly rank-Berkeley cardinals. This connection is analogous to the relationship between certain large cardinals compatible with V=L and their stronger counterparts that contradict it.
Here's a breakdown of the relationship:

Rank-Berkeley cardinals are defined in ZF (without Choice) and necessitate the existence of non-trivial elementary embeddings from $V_ζ$ to itself for arbitrarily large $ζ$. This strong property directly contradicts the Axiom of Choice.

Exacting cardinals can be viewed as a "weakened" version of rank-Berkeley cardinals. Instead of demanding embeddings from the entire $V_ζ$, their definition only requires embeddings from elementary submodels of $V_ζ$ that contain a certain structure. This weakening allows them to be compatible with the Axiom of Choice.

Ultraexacting cardinals further extend the notion of exacting cardinals by imposing an additional condition on the embeddings, requiring them to contain the restriction of the embedding to $V_λ$ in their domain.
The analogy with V=L compatible large cardinals is as follows:

Strongly unfoldable cardinals are a weakening of supercompact cardinals, achieved by restricting the structure of the target model of the embedding. This weakening makes them consistent with V=L.
Similarly, exacting cardinals are a weakening of rank-Berkeley cardinals, achieved by restricting the domain of the embedding. This weakening makes them consistent with the Axiom of Choice.
Reinhardt cardinals, which directly imply the existence of an elementary embedding $j: V \rightarrow V$, represent a stronger notion than rank-Berkeley cardinals. While the paper doesn't explicitly draw a connection between exacting/ultraexacting cardinals and Reinhardt cardinals, the analogy with weaker large cardinals suggests that even stronger large cardinal notions, incompatible with ZFC, might exist and have exacting/ultraexacting cardinals as their "weakened" counterparts.

Could the existence of exacting cardinals be established from weaker large cardinal assumptions than I0?

While the paper demonstrates that the existence of exacting cardinals is consistent relative to I0, it remains an open question whether this assumption can be weakened.
The authors don't provide a definitive answer, but the text hints at potential difficulties in weakening the assumption significantly. For instance, it's mentioned that the existence of a 1-exact embedding at λ implies that there are no extendible cardinals below λ. This suggests that exacting cardinals possess significant strength even without additional large cardinal assumptions.
Furthermore, the proof of the consistency of exacting cardinals from I0 heavily relies on the structure provided by the I0 embedding. It's unclear whether a similar construction could be carried out using weaker large cardinal principles.
Therefore, while it's possible that the consistency of exacting cardinals could be established from weaker assumptions, it seems unlikely that the assumption could be weakened substantially. Further research is needed to explore this question further.

What are the philosophical implications of large cardinals like exacting cardinals, which suggest a fundamental tension between the Axiom of Choice and the principle of definability?

The existence of exacting cardinals, particularly their incompatibility with V=HOD, points towards a fascinating and potentially profound tension between the Axiom of Choice and the principle of definability in set theory. This tension challenges the common intuition that strong axioms of infinity should be compatible with a highly structured and definable universe.
Here's a breakdown of the philosophical implications:

Challenge to Definabilism: The principle of definability, often associated with V=HOD, posits that the universe of sets is ultimately built from definable objects. Exacting cardinals contradict this principle by implying V≠HOD, demonstrating that some sets are inherently undefinable even when allowing parameters from a substantial part of the universe ($V_λ$).

Choice vs. Structure: The Axiom of Choice, while enabling elegant mathematical constructions, often introduces a degree of "arbitrariness" by allowing for choices without explicit definitions. Exacting cardinals, being consistent with Choice but not with V=HOD, suggest that the freedom granted by Choice might be inherently at odds with a highly structured and definable universe.

Limits of Reflection: Large cardinal axioms are often seen as expressions of the "reflection principle," which states that the universe of sets is rich and diverse enough to reflect its properties in smaller structures. Exacting cardinals, however, challenge this view by exhibiting a form of reflection (through the existence of elementary embeddings) that ultimately leads to the existence of undefinable sets.
These implications raise fundamental questions about the nature of the set-theoretic universe:

Is the universe inherently definable or does it contain essential elements of undefinability?
To what extent does the Axiom of Choice contribute to the undefinability present in the universe?
What are the limits of reflection principles in set theory, and how do they interact with the tension between choice and definability?
Exacting cardinals, by highlighting this tension, push the boundaries of our understanding of the interplay between fundamental set-theoretic principles. They suggest that the universe of sets might be far more complex and nuanced than initially envisioned, harboring a delicate balance between structure, definability, and the freedom of choice.