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:
“If I were as I once was, the strong hoofs crushing the sand and the shells,
Coming out of the sea as the dawn comes, a chaunt of love on my lips,
Not coughing, my head on my knees, and praying, and wroth with the bells,
I would leave no saint’s head on his body from Rachlin to Bera of ships.”
—William Butler Yeats (1865–1939)
“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)