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

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

1. π : M → Hθ 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 = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than "

such that in some forcing. Although the definition is similar to one of the definitions of supercompact cardinals.

See also

• Hereditarily countable set

References