Abstract
For a filter F let cF(α) be the cardinality of the set of all filters isomorphic to F on a cardinal α. We derive formulas for these functions similar to cardinal exponential formulas.
We show that precise values of the function cF depends on the filter F and also on the axioms of set theory. We apply these results to get a description of the function bF for a set functor F (bF(α) is the cardinality of F α for a cardinal α).
We prove that the function bF depends on the functor F and on the axioms of set theory. For a partial cardinal function d, we find a sufficient condition for the existence of a set functor F with d(α)=bF(α) for all cardinals α such that d(α) is defined.
We prove that a functor F is finitary if and only if there exists a cardinal β such that bF(α) is less or equal to α for every cardinal α greater or equal to 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.