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Equivalently, an ultrafilter F on a set S is a filter on S with the additional property that for every subset A of S, either A is in F or S\A is in F. |
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Equivalently, an ultrafilter F on a set S is a filter on S with the additional property that for every subset A of S, either A is in F or S \ A is in F. |
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The existence of free ultrafilters cannot be proved without the axiom of choice, and therefore explicit examples of such ultrafilters cannot be given. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter on a finite set is principal. |
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One can show that every filter is contained in an ultrafilter (see Ultrafilter Lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice in the form of Zorn's Lemma, so explicit examples of free ultrafilters cannot be given. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter on a finite set is principal. |
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Ultrafilters are also used in the construction of hyperreal numbers. |
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Ultrafilters are also used in the construction of hyperreal numbers. Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter. |
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