Filter: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above measurable cardinals, although they also have some relation to indescribable and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A...") |
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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 cardinal \(\gamma\) and set \(X\), a filter \(F\) on \([X]^{< \gamma}\) is called fine if, for each \(x \in X\), \(\{\sigma \in [X]^{< \gamma}: x \in \sigma\} \in F\). Similarly, an ultrafilter on a cardinal \(\kappa\) is called fine (or uniform) if, for each \(\alpha < \kappa\), \(\{\sigma
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