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:
“Where might is, the right is:
Long purses make strong swords.
Let weakness learn meekness:
God save the House of Lords!”
—A.C. (Algernon Charles)
“The Cardinal is at his wit’s end—it is true that he had not far to go.”
—George Gordon Noel Byron (1788–1824)