Cofinalities of Borel ideals



We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented math formula ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant math formula and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.