Berkeley Cardinal Hierarchy
Berkeley cardinals and their variations represent the conceptual "ceiling" of non-Axiom of Choice (non-AC) set theory. These cardinals are so powerful that they are inconsistent with the full Axiom of Choice, meaning they can only exist in Zermelo-Fraenkel (ZF) set theory. Their strength is derived from the level of reflection they enforce through elementary embeddings.
In simple terms, a "stronger" variation in this list requires the elementary embeddings to be more "precise," to handle a more complex class of structures, or to impose tighter restrictions on the critical point.
Here is the hierarchy, ordered from the lowest consistency strength to the highest.
1. Virtually-proto-Berkeley (The Base)
This is the weakest definition because it utilizes generic elementary embeddings. In set-theoretic terms, a "virtual" or "generic" property often implies that the necessary mathematical objects exist in a forcing extension of the mathematical universe, rather than in the original universe itself. Because it doesn't require the embedding to exist in the standard model, its consistency strength is significantly lower, making it a foundation from which other, stronger definitions are built.
2. ζ-proto-Berkeley (for some ordinal ζ < δ)
This variation is a localized version of the proto-Berkeley property. It specifically requires an embedding with a critical point that falls within a restricted range, bounded below by a specific ordinal ζ and above by the cardinal δ. It is weaker than the proto-Berkeley variation because it only demands the existence of an embedding for a specific localized interval, rather than for any ordinal threshold below the cardinal.
3. Proto-Berkeley
This serves as the foundational definition for the standard Berkeley hierarchy. A cardinal κ is considered proto-Berkeley if every transitive set M containing κ has at least one elementary embedding j: M ≺ M with a critical point below κ. It is the absolute baseline requirement for the "true" non-AC large cardinal properties.
4. Berkeley
The standard Berkeley cardinal definition is a significant step up from proto-Berkeley. It adds a "density" requirement: for any arbitrary ordinal α < κ, there must exist a transitive set M with κ ∈ M and an elementary embedding j: M ≺ M such that the critical point is specifically between α and κ. This ensures that the embeddings are "dense" below κ, providing much stronger combinatorial reflection.
5. Club Berkeley
This variation requires the critical point of the embedding to fall within any arbitrary club set (a set that is both closed and unbounded) C ⊆ κ. In large cardinal theory, club sets are considered "large" and structurally representative of the cardinal's skeleton. This is a far more restrictive and precise condition on the embedding than the standard Berkeley definition, making it considerably stronger.
6. Limit club Berkeley
By definition, this is a Club Berkeley cardinal that is also a limit of Berkeley cardinals. In large cardinal hierarchies, being a "limit" of a property always places that object higher in the consistency strength, as it implies the existence of an entire sequence of Berkeley cardinals below it.
7. ζ-proto-rank-Berkeley (for some ordinal ζ < δ)
This marks a definitive shift, moving the focus from general transitive sets to the cumulative rank hierarchy Vλ. The rank versions are inherently stronger as Vλ structures are more structured and rigid. This localized version specifies the critical point range and requires δ to be a fixed point of the embedding.
8. Proto-rank-Berkeley
This is the baseline for the rank hierarchy context. Similar to proto-Berkeley, it doesn't have the localized ζ restriction on the critical point, just that it is below δ. However, working in the Vλ context and requiring δ to be a fixed point is much stronger than the basic proto-Berkeley requirement.
9. Rank-Berkeley
This is the rank-to-rank equivalent of the standard Berkeley property. It combines the full range density requirement with the immensely powerful structural tools of rank-to-rank embeddings, requiring embeddings for all λ > δ with critical points density conditions and a fixed point. It combines the best of both worlds.
10. X-closed rank Berkeley (The Strongest)
This is the most powerful large cardinal property listed. In addition to meeting all the criteria for a rank-Berkeley cardinal, it imposes a specific and rigid "pointwise image" or closure property. The embedding must satisfy j(x) = j[x] for all x ∈ X, where X is a specific set. This ensures the embedding is so strong that it perfectly preserves the complex internal membership structure of the set X, a level of precision that places it at the very top of this hierarchy and near the absolute limit of non-AC set-theoretic strength.