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:
“It has been so written, for the most part, that the times it describes are with remarkable propriety called dark ages. They are dark, as one has observed, because we are so in the dark about them.”
—Henry David Thoreau (1817–1862)
“Time and I against any two.”
—Spanish proverb.
Quoted by Cardinal Mazarin during the minority of Louis XIV.