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 → Hθ 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 = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ" Equivalently, κ {\displaystyle \kappa } is remarkable if and only if for every λ > κ {\displaystyle \lambda >\kappa } there is λ ¯ < κ {\displaystyle {\bar {\lambda }}<\kappa } such that in some forcing extension V [ G ] {\displaystyle V[G]} , there is an elementary embedding j : V λ ¯ V → V λ V {\displaystyle j:V_{\bar {\lambda }}^{V}\rightarrow V_{\lambda }^{V}} satisfying j ( crit ( j ) ) = κ {\displaystyle 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 ] {\displaystyle V[G]} , not in V {\displaystyle V} .
See also Hereditarily countable set
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