In mathematics, a remarkable cardinal is a certain kind of large cardinal number.

Formally, a cardinal κ is called remarkable iff for all regular cardinals θ > κ, there exist π, M, λ, σ, N and ρ such that

1. π : M → H_θ is an elementary embedding
2. M is countable and transitive
3. π(λ) = κ
4. σ : M → N is an elementary embedding with critical point λ
5. N is countable and transitive
6. ρ = M ∩ Ord is a regular cardinal in N
7. σ(λ) > ρ
8. M = H_ρ^N, i.e., M ∈ N and N |= "M is the set of all sets that are hereditarily smaller than ρ"

References

• Schindler, Ralf: Proper forcing and remarkable cardinals, Bulletin of Symbolic Logic 6, 2000, pp. 176-184 [1]