# ⓘ 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 card ..

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

Equivalently, κ {\displaystyle \kappa } is remarkable if and only if for every λ > κ {\displaystyle \lambda > \kappa } there is λ ¯ < κ {\displaystyle {\bar {\lambda }}

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