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:
““Would you—be good enough—” Alice panted out, after running a little further, “to stop a minute—just to get—one’s breath again?”
“I’m good enough,” the King said, “only I’m not strong enough. You see, a minute goes by so fearfully quick. You might as well try to stop a Bandersnatch!””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Distrust all those who love you extremely upon a very slight acquaintance, and without any visible reason. Be upon your guard, too, against those who confess, as their weaknesses, all the cardinal virtues.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)