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, is remarkable if and only if for every there is such that in some forcing extension , there is an elementary embedding satisfying . Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in , not in .
See also Hereditarily countable set
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