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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 Hereditarily countable set References Schindler, Ralf (2000), "Proper forcing and remarkable cardinals", The Bulletin of Symbolic Logic 6 (2): 176–184, doi:10.2307/421205, MR1765054, ISSN 1079-8986, http://www.math.ucla.edu/~asl/bsl/0602/0602-003.ps  This set theory-related article is a stub. You can help Wikipedia by expanding it. v • d • e