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:
“To-night he noticed how the women’s eyes
Passed from him to the strong men that were whole.
How cold and late it is! Why don’t they come
And put him into bed? Why don’t they come?”
—Wilfred Owen (1893–1918)
“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)