Ultraexacting Cardinals

• Ultraexacting cardinals are large cardinal axioms characterized by specific nontrivial elementary embeddings with an internal restriction property and definability via Ordinal Definable predicates.
• They are equiconsistent with the I₀ axiom, amplifying consistency strength and reshaping the traditional hierarchical landscape between V and HOD.
• Their construction using I₀ embeddings and forcing methods provides new insights into inner model theory, structural reflection, and the boundaries of large cardinal properties.

Ultraexacting cardinals are a recently introduced class of large cardinal axioms in set theory, distinguished by their relationship to elementary embeddings, structural reflection principles, and definability conditions involving Ordinal Definable (OD) predicates. Unlike classical large cardinal properties, ultraexacting cardinals are compatible with ZFC and have unique consistency and definability consequences for the set-theoretic universe, particularly for the relationship between V and HOD (the class of hereditarily ordinal-definable sets). Their existence is equiconsistent with the axiom I₀ and they are pivotal in the analysis of the structure and boundaries of the large cardinal hierarchy.

1. Formal Definition and Elementary Embeddings

Ultraexacting cardinals are defined via the existence of specific nontrivial elementary embeddings. Let λ be a limit cardinal, and consider an elementary submodel $X \subseteq V_\zeta$ (for suitable $\zeta \in C^{(n+1)}$) satisfying

• $V_\lambda \cup \{\lambda\} \subseteq X$,
• The embedding $j: X \to V_\zeta$ fixes λ ($j(\lambda)=\lambda$),
• The restriction $j|\lambda \neq id_\lambda$ is nontrivial.

The ultraexacting enhancement demands that the restriction $j|V_\lambda$ is itself an element of $X$. This additional "internal" property enables equivalences with structural reflection principles involving "square root" embeddings: given an embedding $j$, a map $r$ exists such that an extension $r^+$ recovers $j$, aligning with literature on rank-into-rank embeddings and square root phenomena.

Equivalently, using OD predicates, λ is ultraexacting if for every ordinal definable $A \subseteq V_{\lambda+1}$ there is an elementary embedding

$j : (V_{\lambda+1}, A) \to (V_{\lambda+1}, A)$

with critical point below λ.

2. Consistency Strength and Hierarchical Position

Ultraexacting cardinals are equiconsistent with the I₀ axiom, one of the strongest classical large cardinal hypotheses. Specifically,

$$\text{ZFC} + \text{``there exists an I}_0\text{-embedding}~j : L(V_{\lambda+1}) \to L(V_{\lambda+1})~\text{with}~j|\lambda \neq \text{Id}_\lambda \text{''}$$

is equiconsistent with

$$\text{ZFC} + \text{``there exists an ultraexacting cardinal''}$$

(Aguilera et al., 12 Sep 2025, Aguilera et al., 18 Nov 2024).

Ultraexacting cardinals "amplify" the consistency strength, particularly when situated with other large cardinal axioms. The presence of an ultraexacting cardinal below a measurable cardinal implies consistency with a proper class of I₀ embeddings. In contrast, exacting cardinals (without the ultracondition) have a consistency strength strictly between I₃ and I₂:

$I_2 \to I_3_1 \to I_3_{wf(n+1)} \to I_3_{wf(0)} \to \text{exacting cardinal} \to I_3$

This hierarchy documents the precise placement inside the large cardinal spectrum, challenging the classical linear-incremental progression.

3. Structural Reflection and Ordinal-Definable Characterizations

Ultraexacting cardinals have multiple equivalent reformulations in terms of reflection and definability:

• Structural Reflection: The principle termed “Ultraexact Structural Reflection” (UXSR) postulates the existence of embeddings between structures extended by predicates or relations, where the witnessing embedding is a square root of a given global map and reflects between structures $\langle \mu, \lambda \rangle$ and $\langle \nu, \lambda \rangle$ (with $\nu < \mu$).
• Ordinal Definability: For every OD subset $A \subseteq V_{\lambda+1}$, there is an elementary embedding $j:(V_{\lambda+1},A)\to(V_{\lambda+1},A)$. This removes reference to ambient elementary substructures and is salient in choiceless inner models.

A two-cardinal “strong unfoldability” characterization is also given: for $n \geq 2$, λ is exacting if $\exists \mu < \lambda$ such that for every suitable $\Sigma_n$-definable class of structures, every member $B$ of type $\langle \mu, \lambda \rangle$ has a substructure $A$ of type $\langle \nu, \lambda \rangle$ with $\nu < \mu$ and $A$ embeddable into $B$.

4. Impact on HOD and Vopěnka’s Principle

The existence of ultraexacting (or exacting) cardinals entails that $V \ne \text{HOD}$. Such strong non-definability consequences do not occur for traditional large cardinals, which may coexist with $V = \text{HOD}$. Notably,

• Exacting cardinals imply that HOD miscalculates the regularity properties of cardinals, e.g., λ can be regular in HOD but singular in $V$ or vice versa (Aguilera et al., 18 Nov 2024).
• If a 2-ultraexact embedding at λ exists, then λ⁺ is ω-strongly measurable in HOD; HOD cannot partition the set of countable cofinality ordinals below λ⁺ into δ many stationary sets.

Regarding global set-theoretic principles:

• The presence of an ultraexacting or exacting cardinal above an extendible cardinal refutes Woodin’s HOD Conjecture and Ultimate–L Conjecture.
• However, the existence of an extendible cardinal above an exacting cardinal does not necessarily refute the HOD Hypothesis; models can be constructed with both properties (Aguilera et al., 12 Sep 2025).

Ultraexacting cardinals also interact with Vopěnka’s Principle: a model with an exacting cardinal and an I₂-embedding is consistent with both Vopěnka’s Principle and the HOD Hypothesis.

5. Relationships with Other Large Cardinal Notions

Ultraexacting cardinals are natural strengthenings of exacting cardinals and relate to several classical and recent notions:

• Weak forms of rank-Berkeley cardinals: By confining the embedding domain to small elementary submodels, ultraexacting cardinals generalize properties formerly confined to the universe or large segments.
• Strong forms of Jónsson cardinals: They guarantee that every definable structure in a countable language reflects to proper elementary substructures, with extra internal properties captured by the exact embeddings.
• α-iterable cardinals: The strict hierarchy of α-iterable cardinals, wherein higher iterability (for larger α) simulates “ultraexacting” behavior for ultrapowers, supports the stratification of large cardinal strength below measurability (Gitman et al., 2011).
• Entities such as C⁽ⁿ⁾-cardinals correspond to refined iterability and correctness properties; requiring embeddings to send critical points to ordinals in $C^{(n)}$ produces a more granular hierarchy (Bagaria, 2019).

6. Methods of Construction and Forcing

Inner model theory and forcing play a central role in realizing ultraexacting cardinals:

• From I₀ Embeddings: Given an I₀ embedding $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$, one can construct in a set-sized model a cardinal that is ultraexacting. If the model contains suitable fine-structural premice, the consistency strength is amplified accordingly (Aguilera et al., 12 Sep 2025, Aguilera et al., 18 Nov 2024).
• Prikry Forcing: Applying Prikry forcing to an I₃_{wf(0)} cardinal can yield a model in which the critical point is exacting, though the blow-up behavior typical of ultraexacting cardinals does not occur for exacting cardinals alone.
• Forcing for Generalizations: Woodin’s definable partial orders (homogeneous, $\Sigma_3$-definable) are used to force models in which every natural number $n$ admits an $n$-ultraexact embedding at λ (thus λ is ultraexacting).
• Inner Model Applications: When combined with countably iterable Mitchell–Steel premice, the existence of ultraexacting cardinals is equiconsistent with self-embeddings of $L(V_{\lambda+1}, M)$ into itself.

7. Consequences and Open Directions

Ultraexacting cardinals have several fundamental consequences:

• They force $V \ne$ HOD, introducing genuine nondefinability phenomena that contrast with the compatibility scenarios of classical large cardinals.
• The interaction between ultraexacting and traditional large cardinals may result in consistency amplification—e.g., a proper class of I₀ embeddings from a single ultraexacting below a measurable.
• Their novel characterizations in terms of OD predicates and reflection principles suggest new lines of research in inner model theory and absoluteness.
• The boundary between ultraexacting properties and “superdestructible” behavior under forcing (as for superstrong or huge cardinals (Bagaria et al., 2013)) clarifies which large cardinal notions survive generic extensions, supporting a nuanced large cardinal hierarchy rather than a linear one.

In sum, ultraexacting cardinals provide a natural, robust, and fine-grained extension of large cardinal theory, equiconsistent with the most potent classical axioms, with significant ramifications for structural and definability properties of the set-theoretic universe, inner model theory, and the hierarchy of reflection principles.