Math Conjectures

Reinhardt Cardinal Inconsistency Problem

OPENMajorOpen problemProposed 1967 · Standard version

Canonical statement

Prove in ZF\mathsf{ZF}, without the axiom of choice, that there is no nontrivial elementary embedding j:V→Vj:V\to V. Equivalently, determine whether the existence of a Reinhardt cardinal, the critical point of such an embedding, is inconsistent with ZF\mathsf{ZF}.
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Prove in \(\mathsf{ZF}\), without the axiom of choice, that there is no nontrivial elementary embedding \(j:V\to V\). Equivalently, determine whether the existence of a Reinhardt cardinal, the critical point of such an embedding, is inconsistent with \(\mathsf{ZF}\).
LOG-SET-009:[McCallum2026Reinhardt][Kunen1971Inconsistency][Kanamori2003HigherInfinite]
Logic and foundations

Notes

Kunen proved with the axiom of choice that there is no nontrivial elementary embedding j:V→Vj:V\to V [Kunen1971Inconsistency]. Whether choice can be removed is the Reinhardt-cardinal inconsistency problem, a standard boundary of the large-cardinal hierarchy [Kanamori2003HigherInfinite]. The current 2026 manuscript claim [McCallum2026Reinhardt] omits material reflection, branch, model, and elementarity steps, so it is linked as a Grade D claim rather than accepted as a theorem.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.