Totally Reinhardt Cardinal
The Reinhardt cardinal represents the conceptual frontier where the paradigm of elementary embeddings reaches its logical absolute: a non-trivial mapping of the entire set-theoretic universe into itself (j: V → V). Because the Kunen Inconsistency Theorem established that such an embedding cannot coexist with the Axiom of Choice, these entities are studied exclusively within Zermelo-Fraenkel set theory without Choice (ZF) or its second-order extension (ZF₂). When mathematicians push this global reflection paradigm to its absolute theoretical limit—demanding that the universe not only reflect onto itself, but do so with structural immunity to any arbitrary class-sized parameter—they arrive at the totally Reinhardt cardinal.
The totally Reinhardt cardinal represents the ultimate conclusion of the Vopěnka reflection hierarchy applied to global universe-level embeddings. It is an infinity so structurally dominant that its localized rank stages possess complete, self-contained knowledge of even stronger variations of global embeddings.
1. The Step-by-Step Formal Architecture
To comprehend the structural configuration of a totally Reinhardt cardinal, one must trace a strict architectural progression of increasing embedding requirements within a choiceless universe.
The Baseline Reinhardt Cardinal
A cardinal κ is Reinhardt if it is the critical point (the first shifted ordinal) of a non-trivial elementary embedding j: V → V. It maps the entire universe into itself, ensuring complete preservation of all first-order mathematical truths.
The Super Reinhardt Cardinal
A cardinal κ is super Reinhardt if it can serve as the critical point for a family of universe embeddings j: V → V where the target image of the critical point, j(κ), can be pushed as high into the ordinals as desired. For any arbitrary ordinal λ, there exists an embedding witnessing that κ is Reinhardt such that j(κ) > λ.
The A-Super Reinhardt Cardinal
To introduce parameters into this global mapping, let A be a proper class. A cardinal κ is A-super Reinhardt if, for every ordinal λ, there is a non-trivial elementary embedding j: V → V such that:
- The critical point is κ (crit(j) = κ).
- The target image exceeds the designated ordinal (j(κ) > λ).
- The embedding perfectly preserves the class A across the entire ordinal hierarchy, meaning j⁺(A) = A (where j⁺ represents the union of the embedding's effects on all localized intersections of A with rank stages).
The Definition of Totally Reinhardt
With these foundational layers established, the definition of total Reinhardtness shifts from the global universe down into a localized, two-tiered second-order structure. A cardinal κ is totally Reinhardt if, for every subset or parameter A belonging to the next power-set layer V_κ₊₁, the two-tiered model (V_κ, V_κ₊₁) satisfies the axioms of second-order set theory (ZF₂) along with the assertion:
"There exists an A-super Reinhardt cardinal."
2. The Ultimate Conclusion of the Vopěnka Hierarchy
The profound depth of a totally Reinhardt cardinal lies in its relationship to structural reflection. In standard ZFC set theory, Vopěnka's Principle asserts a global "crowdedness" of mathematical structures, forcing smaller subsets to reflect embeddings smoothly. Beyond the boundaries of Choice, this principle undergoes a radical evolution.
A totally Reinhardt cardinal represents the ultimate ceiling of this upward reflection. By demanding that the second-order model (V_κ, V_κ₊₁) models the existence of an A-super Reinhardt cardinal for any arbitrary parameter A, the cardinal κ forces its internal ecosystem to be a flawless mirror of hyper-infinite reflection.
No matter how complex a parameter A you select from the power set of V_κ, the structure below κ contains a specialized global embedding capable of absorbing, preserving, and reflecting that exact parameter. It transforms the localized rank stage V_κ into a perfect microcosm of the absolute choiceless universe.
3. Consistency Strength
In the upper limits of large cardinal theory, the totally Reinhardt cardinal sits firmly above standard Reinhardt and super Reinhardt variations. It represents the final boundary of embeddings defined via j: V → V before the hierarchy transitions into the even more aggressive, structural forcing of Berkeley cardinals.
| Cardinal Stratum | Underlying Logical Framework | Definitive Embedding / Model Constraint |
| Reinhardt Cardinal | ZF / ZF₂ (Choiceless) | A single global elementary embedding j: V → V with critical point κ. |
| Super Reinhardt Cardinal | ZF / ZF₂ (Choiceless) | A family of embeddings j: V → V where the critical image j(κ) can exceed any target ordinal λ. |
| Totally Reinhardt Cardinal | ZF₂ (Choiceless) | The localized model (V_κ, V_κ₊₁) satisfies ZF₂ + the existence of an A-super Reinhardt cardinal for all parameters A. |
| Berkeley Cardinal | ZF (Choiceless) | For every transitive set M containing κ, there is a non-trivial embedding j: M → M with crit(j) < κ. Completely dominates all Reinhardt variants. |
Ultimately, the totally Reinhardt cardinal demonstrates that even when the Axiom of Choice is stripped away, the upper limit of set theory does not dissolve into chaos. Instead, it uncovers a landscape of absolute structural symmetry, where a single cardinal can force an entire localized rank stage to echo the infinite capacity of the global universe.