Remarkable cardinal
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In mathematics, a remarkable cardinal is a certain kind of large cardinal number.
Formally, a cardinal κ is called remarkable iff 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"
References
- Schindler, Ralf: Proper forcing and remarkable cardinals, Bulletin of Symbolic Logic 6, 2000, pp. 176-184 [1] (http://www.math.ucla.edu/~asl/bsl/0602/0602-003.ps)