title: Remarkable cardinal
text: # 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 $V$.
  
  