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)
“Honest towards ourselves and towards anyone else who is our friend; brave towards the enemy; magnanimous towards the defeated; polite—always: this is how the four cardinal virtues want us to act.”
—Friedrich Nietzsche (1844–1900)