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:
“Let us pardon him his hope of a vain apocalypse, and of a second coming in great triumph upon the clouds of heaven. Perhaps these were the errors of others rather than his own; and if it be true that he himself shared the general illusion, what matters it, since his dream rendered him strong against death, and sustained him in a struggle to which he might otherwise have been unequal?”
—Ernest Renan (1823–1892)
“Time and I against any two.”
—Spanish proverb.
Quoted by Cardinal Mazarin during the minority of Louis XIV.