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(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 ρ"

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