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:
“In using the strong hand, as now compelled to do, the government has a difficult duty to perform. At the very best, it will by turns do both too little and too much. It can properly have no motive of revenge, no purpose to punish merely for punishment’s sake. While we must, by all available means, prevent the overthrow of the government, we should avoid planting and cultivating too many thorns in the bosom of society.”
—Abraham Lincoln (1809–1865)
“One must not make oneself cheap here—that is a cardinal point—or else one is done. Whoever is most impertinent has the best chance.”
—Wolfgang Amadeus Mozart (1756–1791)