Bagaria-Koellner Axioms
The Bagaria–Koellner Axioms is a family of extraordinarily strong large-cardinal principles. In the technical literature, these principles are more commonly discussed under the designation large cardinals beyond the Axiom of Choice.
Their importance lies in the fact that they investigate elementary embeddings and cardinal principles so powerful that they cannot coexist with the ordinary Axiom of Choice. Instead of merely enlarging the conventional hierarchy of large cardinals within ZFC, these principles ask what becomes possible when set theory is studied in ZF, where the Axiom of Choice is not assumed.
The resulting hierarchy includes Reinhardt cardinals, super Reinhardt cardinals, totally Reinhardt cardinals, proto-Berkeley cardinals, Berkeley cardinals, club Berkeley cardinals, and limit club Berkeley cardinals. These principles represent some of the strongest attempts to describe elementary self-reflection within the cumulative hierarchy of sets.
These axioms are not simply assertions that exceptionally large collections exist. Their defining feature is the existence of highly structured elementary embeddings. An elementary embedding is a mapping between set-theoretic structures that preserves every statement expressible in the first-order language of set theory.
When an embedding j: M → N is elementary, the structure M and its image inside N agree about the truth of every first-order statement concerning the objects being mapped. The embedding may move certain sets and ordinals while preserving the complete logical organization of the source structure.
This makes elementary embeddings among the strongest available instruments for describing profound self-similarity, reflection, and structural recurrence within the cumulative hierarchy.
Elementary Embeddings and Critical Points
The central object appearing throughout the Bagaria–Koellner framework is a nontrivial elementary embedding of the following kind:
j: M → M
In the Reinhardt context, the embedding has the stronger global expression:
j: V → V
The symbol V represents the complete cumulative hierarchy of sets. An embedding is called nontrivial when it does not leave every object unchanged.
The least ordinal moved by j is known as the critical point of the embedding. It is written as:
crit(j)
If κ is the critical point of j, then every ordinal below κ remains fixed:
j(α) = α for every α < κ
At κ itself, the embedding begins to move the ordinal structure upward:
j(κ) > κ
Therefore:
κ = crit(j)
The critical point is not merely a large cardinal in the ordinary numerical sense. It is the threshold at which an elementary structural transformation first becomes nontrivial. The cardinal records the beginning of a self-embedding that preserves the complete first-order organization of the structure while shifting part of its ordinal architecture.
This is what distinguishes embedding-based large cardinals from simpler assertions of largeness. An inaccessible cardinal, for example, is defined through closure and regularity properties. A measurable cardinal is associated with a highly complete ultrafilter. The cardinals considered within the Bagaria–Koellner hierarchy are characterized through increasingly comprehensive patterns of elementary self-embedding.
The strength of one of these principles depends not only upon the existence of an embedding, but also upon where embeddings must exist, how their critical points are distributed, which structures must admit them, and how extensively those embeddings reflect the large-cardinal hierarchy beneath themselves.
Why the Axiom of Choice Matters
The Axiom of Choice, abbreviated as AC, states that for every collection of nonempty sets, there exists a function selecting one element from each set. Although this principle may initially appear modest, it has far-reaching consequences throughout set theory.
ZF together with the Axiom of Choice is denoted ZFC. Set theory formulated without assuming Choice is denoted ZF. Without Choice, collections need not necessarily admit well-orderings, cardinal comparison becomes more subtle, and structures forbidden within ZFC may become conceivable.
Under the conventional Choice-based framework, there can be no nontrivial elementary embedding:
j: V → V
This result is commonly known as the Kunen Inconsistency Theorem. Consequently, Reinhardt cardinals and stronger principles developed above them cannot exist within ordinary ZFC.
The Bagaria–Koellner–Woodin program therefore investigates these axioms outside the jurisdiction of Choice. It asks whether the obstruction arises from elementary embeddings themselves or specifically from their interaction with the global well-ordering consequences supplied by the Axiom of Choice.
In this way, the program examines not only stronger large cardinals, but also the precise structural cost of Choice. It considers whether abandoning global selection principles allows elementary self-reflection to extend beyond the limitations imposed by ZFC.
The phrase beyond Choice should not be interpreted as meaning that these axioms merely ignore Choice. Many of them imply that substantial fragments of Choice must fail. Their existence would require a set-theoretic environment whose organization differs fundamentally from the familiar ZFC setting.
This is why they represent a distinct foundational research program rather than an uncomplicated continuation of the conventional large-cardinal hierarchy.
Reinhardt Cardinals
A cardinal κ is called a Reinhardt cardinal when there exists a nontrivial elementary embedding:
j: V → V
such that:
crit(j) = κ
This means that the complete cumulative hierarchy admits an elementary transformation into itself, with κ as the first ordinal moved by that transformation.
Every set-theoretic truth expressible in the first-order language of membership is preserved through the embedding, even though the embedding reorganizes part of the cardinal and ordinal structure.
The Reinhardt principle is far stronger than the traditional large-cardinal axioms compatible with Choice. Its defining embedding acts upon the entirety of V rather than upon one bounded segment of the cumulative hierarchy.
The total hierarchy is represented within itself through a nontrivial truth-preserving mapping. This does not mean that V becomes a set contained within itself. Rather, the elementary embedding is a proper-class mapping whose reach is unrestricted by any fixed rank of the hierarchy.
Super Reinhardt Cardinals
A super Reinhardt cardinal strengthens the Reinhardt requirement by demanding not merely one elementary embedding with critical point κ, but embeddings capable of moving κ arbitrarily high through the ordinal hierarchy.
A cardinal κ is super Reinhardt when, for every ordinal λ, there exists a nontrivial elementary embedding:
j: V → V
such that:
crit(j) = κ
and:
j(κ) > λ
The decisive phrase is for every ordinal λ. No predetermined ordinal can serve as a ceiling for the image of κ. Given any proposed bound, another elementary embedding exists that carries κ beyond it.
The axiom therefore asserts an inexhaustible family of elementary self-embeddings centered upon the same critical point.
A Reinhardt cardinal guarantees that κ can be moved. A super Reinhardt cardinal guarantees that the movement of κ cannot be bounded by any ordinal. The distinction is therefore qualitative as well as quantitative.
The super Reinhardt condition presents the critical point as the stable origin of arbitrarily extensive elementary transformations of the complete set-theoretic hierarchy.
Totally Reinhardt Cardinals
The notion of a totally Reinhardt cardinal strengthens the hierarchy through second-order reflection and the preservation of additional predicates.
Roughly stated, total Reinhardtness demands that the relevant super Reinhardt behavior persist even when the structure is expanded by arbitrary predicates or subclasses coded below the cardinal.
Ordinary elementarity preserves statements made in the basic language of membership. A predicate-enhanced requirement asks that the embedding also respect supplementary structural information.
This is significant because a mapping that remains elementary after additional predicates are introduced must preserve more than the bare membership relation. It must preserve the enriched organization represented by those predicates.
Total Reinhardtness therefore expresses a more comprehensive invariance. The cardinal is not associated merely with elementary self-embeddings of a minimally described hierarchy. It supports elementary behavior robust enough to survive the inclusion of arbitrary additional structural data available at the relevant level.
Proto-Berkeley Cardinals
The Berkeley hierarchy changes the orientation of the embedding principle. Rather than beginning with one class embedding from V into itself, it examines elementary self-embeddings of every sufficiently inclusive transitive set.
Let E(M) denote the collection of elementary embeddings:
j: M → M
where M is a transitive set.
A set M is transitive when every member of an element of M is itself an element of M. Symbolically, this means:
If x ∈ y and y ∈ M, then x ∈ M.
Transitive sets are especially important because their membership relation correctly reflects the membership structure of the surrounding cumulative hierarchy.
A cardinal δ is a proto-Berkeley cardinal when, for every transitive set M containing δ, there exists a nontrivial elementary embedding:
j: M → M
whose critical point lies below δ:
crit(j) < δ
Thus, every transitive environment large enough to contain δ must possess an elementary self-transformation initiated somewhere beneath δ.
The self-embedding phenomenon is no longer confined to a single global map. It is required throughout every transitive set in which the cardinal appears.
Berkeley Cardinals
A cardinal δ is a Berkeley cardinal when, for every transitive set M containing δ and every ordinal η below δ, there exists a nontrivial elementary embedding:
j: M → M
such that:
η < crit(j) < δ
The essential strengthening over the proto-Berkeley condition is the quantification over every η below δ.
A proto-Berkeley cardinal requires some critical point below δ. A Berkeley cardinal requires suitable critical points arbitrarily close to δ from below. No ordinal beneath δ can permanently bound the locations of the critical points.
This produces an extraordinary density of elementary self-embedding behavior. Every transitive set containing δ must admit nontrivial elementary self-embeddings whose critical points occur unboundedly throughout δ.
Given any lower threshold η, another embedding can be found whose first moved ordinal lies above that threshold while remaining below δ.
A Berkeley cardinal may therefore be understood as a locus of unavoidable elementary self-reflection. Any transitive set capable of containing the cardinal is compelled to exhibit deep internal symmetries, and the points at which those symmetries become nontrivial cannot be confined to a bounded portion of δ.
The Berkeley property is distributed throughout the entire region below the cardinal.
Club Berkeley Cardinals
A club Berkeley cardinal strengthens this distribution requirement by using closed unbounded sets.
A subset C of δ is called closed and unbounded, or a club, when it is unbounded in δ and contains the limits of its increasing sequences whenever those limits remain below δ.
Symbolically:
C ⊆ δ
Club sets represent structurally unavoidable regions of a regular cardinal. They are not merely large by cardinality. They are cofinal in the cardinal and closed under the relevant limiting processes.
A regular cardinal δ is a club Berkeley cardinal when, for every club set C contained in δ and every transitive set M containing δ, there exists an elementary embedding:
j: M → M
such that:
crit(j) ∈ C
The critical points of the available embeddings must therefore intersect every club subset of δ.
They do not merely occur unboundedly below the cardinal. They constitute what set theorists call a stationary pattern: a pattern that cannot be avoided by passing to any closed unbounded region.
This distinction is substantial. An unbounded collection can still avoid a particular club. A stationary collection cannot.
Club Berkeley cardinals therefore require elementary self-embedding activity to be woven into the structural backbone of δ. The possible critical points are unavoidable across every cofinal and limit-closed route through the cardinal.
A club Berkeley cardinal is totally Reinhardt. This connects the locally distributed Berkeley-style embeddings with the far-reaching Reinhardt hierarchy.
The Berkeley and Reinhardt branches are therefore not isolated constructions. At sufficiently strong levels, their reflection principles converge.
Limit Club Berkeley Cardinals
A cardinal δ is a limit club Berkeley cardinal when it is both a club Berkeley cardinal and a limit of Berkeley cardinals.
This means that Berkeley cardinals occur unboundedly below δ. For every ordinal α below δ, there is some Berkeley cardinal β satisfying:
α < β < δ
There is therefore no greatest Berkeley cardinal below δ.
The axiom combines two layers of structural saturation. First, the critical points of elementary self-embeddings must intersect every club subset of δ. Second, the Berkeley property itself must recur unboundedly below δ.
The cardinal does not merely possess an exceptionally strong embedding property at its own level. It stands above an ascending accumulation of cardinals that already possess Berkeley reflection.
This is characteristic of the strongest large-cardinal axioms. Strength is expressed not only by what happens at the cardinal itself, but also by how much of the preceding hierarchy must be reflected, repeated, or accumulated beneath it.
A limit club Berkeley cardinal is therefore a convergence point at which elementary self-embedding behavior has become both structurally unavoidable and recursively abundant.
Reflection Within the Hierarchy
The Bagaria–Koellner hierarchy is governed by the principle of reflection. Reflection means that a property manifested at a very high level is reproduced, approximated, or internally recognized at lower levels.
Strong cardinals frequently imply that weaker large cardinals exist below them, but the axioms beyond Choice exhibit especially forceful versions of this principle.
A super Reinhardt cardinal, for example, reflects Reinhardt-style structure into lower ranks. Berkeley cardinals likewise produce strong internal models containing Reinhardt behavior. Club Berkeley cardinals imply total Reinhardtness. Limit club Berkeley cardinals combine this reflection with an accumulation of Berkeley cardinals below themselves.
The hierarchy is therefore not a simple sequence in which each new axiom merely states that a larger cardinal exists. Each higher principle reorganizes the lower hierarchy by forcing increasingly comprehensive patterns of self-similarity and recurrence.
The Reinhardt branch may be represented schematically as:
Reinhardt → Super Reinhardt → Totally Reinhardt
The Berkeley branch may be represented schematically as:
Proto-Berkeley → Berkeley → Club Berkeley → Limit Club Berkeley
These arrows should not be interpreted as an exhaustive statement of every implication or consistency relationship. The interaction between the Reinhardt and Berkeley branches is more intricate than a single linear ordering.
Nevertheless, the diagrams convey the principal conceptual progression: from the existence of one elementary self-embedding, to arbitrarily extensive embeddings, to embeddings distributed through every transitive set, and finally to critical points that are unavoidable across every club subset of the relevant cardinal.
The Relationship Between Choice and the Least Berkeley Cardinal
One of the most revealing discoveries in this program is the connection between Berkeley cardinals and the failure of Choice.
Suppose δ0 is the least Berkeley cardinal. The cofinality of δ0—the smallest size of an unbounded sequence approaching it—is closely related to which fragments of the Axiom of Choice can remain valid.
If:
γ = cf(δ0)
then particular dependent-choice principles associated with γ cannot remain valid under the corresponding Berkeley assumptions.
The least Berkeley cardinal therefore acts as a measurement of how much Choice the surrounding theory can sustain. The stronger the surviving choice principles are, the more constrained the possible cofinality of the least Berkeley cardinal becomes.
This is philosophically important because it shows that the failure of Choice is not an arbitrary background defect. It is mathematically coordinated with the embedding structure.
The degree to which selections can be made, sequences can be constructed, and collections can be well ordered is connected to the precise architecture of the strongest cardinals.
The axioms beyond Choice therefore transform the absence of AC into a graded and analyzable structural phenomenon.
The HOD Dichotomy
The wider significance of these axioms is connected to HOD, the class of hereditarily ordinal-definable sets.
A set belongs to HOD when it and every object appearing throughout its hereditary membership structure can be defined using ordinals as parameters.
HOD is an internally definable model of ZFC and may be interpreted as the highly organized, ordinal-definable core of the surrounding set-theoretic hierarchy.
Woodin’s HOD Dichotomy states, under an extendible-cardinal hypothesis, that HOD must behave in one of two radically different ways.
Either HOD is structurally close to V, correctly computing major portions of the cardinal hierarchy, or HOD is far from V, with many regular cardinals of V appearing measurable inside HOD.
Bagaria, Koellner, and Woodin presented large cardinals beyond Choice as part of a program investigating the far side of this dichotomy.
This creates a confrontation between two possible pictures of set-theoretic structure. One picture seeks a highly canonical inner model, frequently associated with the Ultimate-L program, in which the underlying hierarchy displays extensive order and definability.
The other picture permits choiceless large cardinals whose existence would undermine central conjectures supporting that canonical interpretation.
The Bagaria–Koellner axioms are consequently not isolated statements about enormous cardinals. They bear directly upon the question of whether the complete set-theoretic hierarchy is fundamentally close to its definable core or radically exceeds it.
Their Foundational Meaning
The deepest meaning of the Bagaria–Koellner axioms concerns the possibility that set-theoretic structure can become self-reflective to a degree prohibited by Choice.
In ordinary large-cardinal theory, stronger axioms provide larger and more comprehensive reflection principles while remaining compatible with global well-ordering.
The axioms beyond Choice cross a threshold at which the demand for elementary self-similarity becomes incompatible with that global organization.
A Reinhardt embedding says that the complete cumulative hierarchy can be elementarily mapped into itself.
A Berkeley cardinal says that every transitive set containing the cardinal must admit elementary self-embeddings with critical points arbitrarily high below it.
A club Berkeley cardinal says that those critical points cannot be avoided by any closed unbounded structural region.
A limit club Berkeley cardinal says that this unavoidable self-reflection is itself accumulated above an unbounded hierarchy of Berkeley cardinals.
The hierarchy therefore describes progressive intensifications of structural recurrence. First, the hierarchy reflects itself. Then it reflects itself through arbitrarily extensive movements. Then every transitive environment containing a particular cardinal is forced to reflect itself.
Finally, the locations initiating those reflections become stationary, unavoidable, and recursively accumulated.
Conclusion
The Bagaria–Koellner Axioms describe one of the most extreme frontiers of contemporary set theory. They investigate cardinals whose defining elementary embeddings are too powerful to coexist with the ordinary Axiom of Choice.
Beginning with Reinhardt cardinals and ascending through super Reinhardt, totally Reinhardt, proto-Berkeley, Berkeley, club Berkeley, and limit club Berkeley principles, the hierarchy progressively strengthens the demand that set-theoretic structures reproduce their own truths through nontrivial elementary self-embeddings.
Their significance cannot be reduced to numerical size. These cardinals represent thresholds of logical preservation, structural self-similarity, reflection, and the controlled failure of Choice.
They ask whether the cumulative hierarchy can contain cardinals around which elementary self-transformation becomes not merely possible, but unavoidable across every transitive setting.
In doing so, they expose a foundational tension between global selection and absolute structural reflection. The more completely the hierarchy is required to preserve and reproduce itself through elementary embeddings, the less compatible it becomes with the universal well-ordering principles supplied by Choice.
The Bagaria–Koellner framework therefore stands as both a large-cardinal hierarchy and a foundational investigation into the possible architectures of set theory.
It examines how far reflection can be extended, how much Choice must be surrendered to permit that extension, and whether the complete hierarchy is ultimately governed by a canonical definable order or by elementary symmetries surpassing the reach of such an order.