Set Functors and Filters

Abstract

For a filter $\mathcal {F}$ let $\mathfrak {c}_{\mathcal {F}}(\alpha )$ be the cardinality of the set of all filters isomorphic to $\mathcal {F}$ on a cardinal α. We derive formulas for these functions similar to cardinal exponential formulas. We show that precise values of the function $\mathfrak {c}_{\mathcal {F}}$ depends on the filter $\mathcal {F}$ and also on the axioms of set theory. We apply these results to get a description of the function $\mathfrak {b}_{F}$ for a set functor F ($\mathfrak {b}_{F}(\alpha )$ is the cardinality of F α for a cardinal α). We prove that the function $\mathfrak {b}_{F}$ depends on the functor F and on the axioms of set theory. For a partial cardinal function $\mathfrak {d}$, we find a sufficient condition for the existence of a set functor F with $\mathfrak {d}(\alpha )=\mathfrak {b}_{F}(\alpha )$ for all cardinals α such that $\mathfrak {d}(\alpha )$ is defined. We prove that a functor F is finitary if and only if there exists a cardinal β such that $\mathfrak {b}_{F}(\alpha )\le \alpha$ for every cardinal α ≥ β. We prove an analogous necessary condition for small set functors and we prove that the precise characterization of small set functors depends on the axioms of set theory.