Filter: Difference between revisions

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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 set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). Note that, if \(X = \bigcup X\), 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!

For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). 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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 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 set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). Note that, if \(X = \bigcup X\), 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!
+For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). 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!

Filter: Difference between revisions

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