Remarkable cardinal


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 = 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 > ρ
  8. M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"
Equivalently, is remarkable if and only if for every there is such that in some forcing extension, there is an elementary embedding satisfying. Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in, not in.