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An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter.
One of the most natural examples of a filter is a principal filter: given some \(x \in X\), the principal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore typically studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\).
It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency.
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