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)
“The Poles do not know how to hate, thank God.”
—Stefan, Cardinal Wyszynski (1901–1981)