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<sub>θ</sub> 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<sub>ρ</sub><sup>N</sup>, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"
Equivalently, <math>\kappa</math> is remarkable if and only if for every <math>\lambda>\kappa</math> there is <math>\bar\lambda<\kappa</math> such that in some forcing extension <math>V[G]</math>, there is an elementary embedding <math>j:V_{\bar\lambda}^V\rightarrow V_\lambda^V</math> satisfying <math>j(\operatorname{crit}(j))=\kappa</math>. Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in <math>V[G]</math>, not in <math>V</math>.
See also
- Hereditarily countable set