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"
Famous quotes containing the words remarkable and/or cardinal:
“Isn’t it remarkable how everyone who knew Lawrence has felt compelled to write about him? Why, he’s had more books written about him than any writer since Byron!”
—Aldous Huxley (1894–1963)
“The Cardinal is at his wit’s end—it is true that he had not far to go.”
—George Gordon Noel Byron (1788–1824)