The completeness of an ultrafilter U on a set is the smallest cardinal κ such that there are κ elements of U whose intersection is not in U. The definition implies that the completeness of any ultrafilter is at least . An ultrafilter whose completeness is greater than —that is, the intersection of any countable collection of elements of U is still in U—is called countably complete or -complete.
The completeness of a countably complete nonprincipal ultrafilter on a set is always a measurable cardinal.
Read more about this topic: Ultrafilter
Famous quotes containing the word completeness:
“Poetry presents indivisible wholes of human consciousness, modified and ordered by the stringent requirements of form. Prose, aiming at a definite and concrete goal, generally suppresses everything inessential to its purpose; poetry, existing only to exhibit itself as an aesthetic object, aims only at completeness and perfection of form.”
—Richard Harter Fogle, U.S. critic, educator. The Imagery of Keats and Shelley, ch. 1, University of North Carolina Press (1949)