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) &gt; ρ
 * 8) M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than &rho;"

Equivalently, $$\kappa$$ is remarkable if and only if for every $$\lambda>\kappa$$ there is $$\bar\lambda<\kappa$$ such that in some forcing extension $$V[G]$$, there is an elementary embedding $$j:V_{\bar\lambda}^V\rightarrow V_\lambda^V$$ satisfying $$j(\operatorname{crit}(j))=\kappa$$. Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in $$V[G]$$, not in $$V$$.