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:
“I confess I was surprised to find that so many men spent their whole day, ay, their whole lives almost, a-fishing. It is remarkable what a serious business men make of getting their dinners, and how universally shiftlessness and a groveling taste take refuge in a merely ant-like industry. Better go without your dinner, I thought, than be thus everlastingly fishing for it like a cormorant. Of course, viewed from the shore, our pursuits in the country appear not a whit less frivolous.”
—Henry David Thoreau (1817–1862)
“One must not make oneself cheap here—that is a cardinal point—or else one is done. Whoever is most impertinent has the best chance.”
—Wolfgang Amadeus Mozart (1756–1791)