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:
“Have We not made the earth as a cradle
and the mountains as pegs?
And We created you in pairs,
and We appointed your sleep for a rest;
and We appointed night for a garment,
and We appointed day for a livelihood.
And We have built above you seven strong ones,
and We appointed a blazing lamp
and have sent down out of the rain-clouds water cascading
that We may bring forth thereby grain and plants,
and gardens luxuriant.”
—Qur’an, “The Tiding” 78:6-16, ed. Arthur J. Arberry (1955)
“The Cardinal is at his wit’s end—it is true that he had not far to go.”
—George Gordon Noel Byron (1788–1824)