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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
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.
For a
Note that, if \(X \subseteq \bigcup X\), then every fine filter is nonprincipal: assume \(F\) is principal, witnessed by \(x \in X\). Then \(x \in \bigcup X\) so, if \(F\) were fine, then \(\{s \in X: x \in s\} \in F\), thus \(x \in x\) - a contradiction!
Similarly, a filter \(F\) on \(X\) is called normal, if, for each function \(f: X \to \bigcup X\), if \(\{s \in X: f(s) \in s\} \in F\), then there is some \(x \in \bigcup X\) so that \(\{s \in X: f(s) = x\} \in F\). This definition is inspired by the [[Fodor's lemma|pressing-down lemma]]. Dually to the fact that no fine filter is principal, every principal filter is normal: assume \(F\) is principal, witnessed by \(x \in X\), and \(f: X \to \bigcup X\). Then, if \(\{s \in X: f(s) \in s\} \in F\), we have \(f(x) \in x\), and so, letting \(x' = f(x)\), we have \(x' \in \bigcup X\), and \(\{s \in X: f(s) = x'\} \in F\). Note, however, that there can be nonprincipal filters which are normal - it is even possible for a filter to be both normal and fine!
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