Large cardinals

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 1. * π : M → Hθ is an elementary embedding 2. * M is countable and transitive 3. * π(λ) = κ 4. * σ : M → N is an elementary embedding with critical point λ 5. * N is countable and transitive 6. * ρ = M ∩ Ord is a regular cardinal in N 7. * σ(λ) > ρ 8. * M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ" Equivalently, is remarkable if and only if for every there is such that in some forcing extension , there is an elementary embedding satisfying . Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in , not in . (Wikipedia).

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Related pages

Large cardinal | Regular cardinal | Supercompact cardinal | Transitive set | Cardinal number | Mathematics | Ordinal number | Hereditarily countable set | Critical point (set theory)