LARGE CARDINAL HIERARCHY
consistency strength ↑
Reinhardt
κ
Reinhardt 1967 / Kunen 1971
j: V → V — inconsistent with AC (Kunen's theorem)
I0
λ
Woodin ~1980s
j: L(Vλ+1) → L(Vλ+1)
I1
λ
Woodin ~1980s
j: Vλ+1 → Vλ+1 — rank-into-rank
I3
λ
Laver ~1980s
j: Vλ → Vλ — nontrivial elementary embedding
Huge
κ
Kunen 1971
j: V → M with M^j(κ) ⊆ M
Extendible
κ
Reinhardt 1970
For every α, j: Vκ+α → Vj(κ)+α
Supercompact
κ
Solovay–Reinhardt 1978
Elementary embedding j: V → M with j(κ) > λ, M^λ ⊆ M
Woodin
δ
Woodin ~1984
For every A ⊆ Vδ, stationary set of strong-for-A cardinals below
Strong
κ
Mitchell 1979
Elementary embedding with Vκ+α in target for each α
Measurable
κ
Ulam 1930
Admits a κ-complete non-principal ultrafilter
Ramsey
κ
Erdős–Hajnal 1962
Every coloring of [κ]<ω has a homogeneous set of size κ
Ineffable
κ
Jensen 1972
Stationary agreement on regressive functions
Subtle
κ
Jensen 1972
Every κ-sequence has a closed unbounded agreement
Indescribable
Π¹ₙ
Hanf–Scott 1961
No first-order property distinguishes it from below
Weakly compact
κ
Erdős–Tarski 1943
Every κ-coloring has a homogeneous set
Mahlo
κ
Mahlo 1911
Set of inaccessibles below is stationary
Inaccessible
κ
Hausdorff 1908
Cannot be reached from below by powerset or replacement
↓ ZFC ↓