In Zermelo-Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal such that every partition of the set of size subsets of into less than pieces has a homogeneous set of size .
The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies that ℵ1 is a strong partition cardinal.
Famous quotes containing the words strong and/or cardinal:
“The enemy is like a woman, weak in face of opposition, but correspondingly strong when not opposed. In a quarrel with a man, it is natural for a woman to lose heart and run away when he faces up to her; on the other hand, if the man begins to be afraid and to give ground, her rage, vindictiveness and fury overflow and know no limit.”
—St. Ignatius Of Loyola (1491–1556)
“Time and I against any two.”
—Spanish proverb.
Quoted by Cardinal Mazarin during the minority of Louis XIV.