A Reinhardt cardinal is one of the most extraordinary large cardinal concepts ever proposed in set theory, representing a level of structural self-similarity that extends beyond virtually every standard large cardinal axiom. Unlike many large cardinal notions that are characterized by elementary embeddings between different structures of the cumulative hierarchy, a Reinhardt cardinal is defined through the existence of a nontrivial elementary embedding of the entire set-theoretic universe into itself. This idea introduces a profound vision of mathematical reality in which the cumulative hierarchy possesses a remarkable degree of internal symmetry, allowing the entire universe of sets to be mapped into itself while preserving every first-order truth. Because of the immense strength of this property, Reinhardt cardinals occupy a unique and highly debated position within the foundations of mathematics.

Formally, a cardinal κ is called a Reinhardt cardinal if it is the critical point of a nontrivial elementary embedding j: V → V, where V denotes the entire cumulative hierarchy of sets. The critical point is the smallest ordinal moved by the embedding, meaning that every ordinal below κ remains fixed while κ itself is mapped to a strictly larger ordinal. Since the embedding is elementary, every first-order statement true in the original universe remains true after the transformation. Consequently, the embedding preserves the complete first-order structure of set theory, making the universe appear, in a precise logical sense, as though it contains an exact image of itself embedded within itself.

The defining feature of a Reinhardt cardinal is not extraordinary size alone but extraordinary structural coherence. Most familiar large cardinal axioms describe enormous stages of the cumulative hierarchy possessing exceptional combinatorial, reflection, or embedding properties. A Reinhardt cardinal instead requires that the universe itself admit an elementary self-embedding. This transforms the cumulative hierarchy into an object exhibiting an exceptional degree of logical self-similarity, where the entirety of set-theoretic reality reflects itself through a nontrivial elementary transformation. Rather than merely extending upward into larger stages, the universe displays an internal architecture capable of reproducing its own logical structure without sacrificing any first-order truths.

One of the most significant aspects of Reinhardt cardinals is their relationship to the Axiom of Choice. The Kunen reality established its inconsistency theorem, establishing that if the standard Zermelo–Fraenkel axioms are combined with the Axiom of Choice (ZFC), then no nontrivial elementary embedding j: V → V can exist. Consequently, Reinhardt cardinals are inconsistent with ZFC. This result, now known as Kunen's Inconsistency Theorem, effectively places Reinhardt cardinals beyond the framework of conventional set theory. Their reality therefore requires weakening or abandoning the Axiom of Choice, leading layers into alternative foundational systems where such embeddings may consistently exist.

Reinhardt cardinals also occupy an important place within the broader hierarchy of embedding-based large cardinals. Many familiar notions—including measurable, strong, supercompact, huge, and extendible cardinals—are characterized by elementary embeddings between different models or different stages of the cumulative hierarchy. Reinhardt cardinals transcend these concepts by replacing embeddings between distinct structures with an elementary embedding of the entire universe into itself. This represents a dramatic conceptual leap, shifting attention from localized reflection phenomena to the global logical architecture of the cumulative hierarchy as a whole.

From a foundational perspective, Reinhardt cardinals invite profound philosophical reflection concerning the nature of mathematical infinity and structural identity. The existence of an elementary self-embedding suggests that the universe of sets may possess forms of internal symmetry far richer than those ordinarily contemplated within classical foundations. Such a possibility challenges traditional assumptions regarding the uniqueness and rigidity of the cumulative hierarchy, raising questions about whether mathematical reality might exhibit recursive or self-referential structural patterns at its highest levels. These ideas have influenced ongoing discussions surrounding elementary embeddings, reflection principles, determinacy, and alternative conceptions of the foundations of mathematics.

Ultimately, the Reinhardt cardinal stands as one of the boldest concepts ever introduced in large cardinal theory. Defined as the critical point of a nontrivial elementary embedding of the entire universe of sets into itself, it represents an unparalleled degree of logical self-similarity and structural preservation. Although incompatible with the Axiom of Choice through Kunen's Inconsistency Theorem, Reinhardt cardinals continue to inspire foundational research within weaker axiomatic systems where their consistency remains unknown. Their significance lies not merely in extending the hierarchy of large cardinals but in fundamentally reimagining the possible architecture of the cumulative universe, making them one of the most profound and enigmatic ideas in all of modern set theory.