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 → 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"
See also
References
- Schindler, Ralf (2000), "Proper forcing and remarkable cardinals", The Bulletin of Symbolic Logic 6 (2): 176–184, doi:10.2307/421205, ISSN 1079-8986, MR 1765054
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