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 unities, sir,’ he said, “are a completeness—a kind of universal dovetailedness with regard to place and time—a sort of general oneness, if I may be allowed to use so strong an expression. I take those to be the dramatic unities, so far as I have been enabled to bestow attention upon them, and I have read much upon the subject, and thought much.””
—Charles Dickens (1812–1870)
“To this war of every man against every man, this also is consequent; that nothing can be Unjust. The notions of Right and Wrong, Justice and Injustice have there no place. Where there is no common Power, there is no Law; where no Law, no Injustice. Force, and Fraud, are in war the two Cardinal virtues.”
—Thomas Hobbes (1579–1688)