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 → 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"

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