Hakimi Cardinal

The Hakimi cardinal is a large cardinal notion proposed by Xuwang Zhihuan in 2026, and its relative consistency with ZFC was later proven.

We define a k-cofinally complete filter over a cardinal k: A filter on k is k-cofinally complete if k has countable cofinality and the filter is σ-complete, or k has uncountable cofinality. For any a < cf(k), the intersection of the family {Xb: b ∈ a} is contained in filter F whenever this family is a subset of F.

A cardinal k is a Hakimi cardinal if there exists a nonprincipal k-cofinally complete ultrafilter on k, alongside an elementary embedding j: L_k → L_2^k.

Lemma (ZFC + A measurable cardinal exists): A Hakimi cardinal exists.

Proof

We adopt this theorem: If a Ramsey cardinal exists, then for any uncountable cardinals k and λ, (Lk, ∈) is an elementary submodel of (Lλ, ∈).

The existence of a measurable cardinal entails a Ramsey cardinal. Meanwhile, every measurable cardinal k has a nonprincipal k-cofinally complete ultrafilter. Since k and 2^k are both uncountable, it follows directly from the above theorem that k is a Hakimi cardinal. This completes the proof.

It is conjectured that the consistency upper bound can be reduced to ZFC plus the existence of a Ramsey cardinal, but a full proof has not yet been completed.