Filter: Difference between revisions

Line 19:

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 \(\alpha < \kappa\), \(\{\sigma ~~< \kappa~~: \alpha < \sigma\} = (\~~sigma,~~\~~kappa)~~ \in F\).

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 \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\).

Revision as of 19:12, 7 September 2023 (edit) 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...") Tag: Visual edit		Revision as of 19:13, 7 September 2023 (edit) (undo) RhubarbJayde (talk \| contribs) No edit summary Tag: Visual edit Newer edit →
Line 19:		Line 19:
	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.		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 \(\alpha < \kappa\), \(\{\sigma ~~< \kappa~~: \alpha < \sigma\} = (\~~sigma,~~\~~kappa)~~ \in F\).		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 \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\).

Filter: Difference between revisions

Navigation menu

Search