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

π : M → H_θ is an elementary embedding
M is countable and transitive
π(λ) = κ
σ : M → N is an elementary embedding with critical point λ
N is countable and transitive
ρ = M ∩ Ord is a regular cardinal in N
σ(λ) > ρ
M = H_ρ^N, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"

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 $V$ .