Cardinal invariants associated with ideals between measure zero and strong measure zero

Abstract

In 2002, Yorioka [Y02] introduced the sigma-ideal I_f for increasing functions f from omega to omega to analyze the cofinality of the strong measure zero ideal. We use and generalize some techiques of itearations with finite support to construct a model of ZFC, by a matrix iteration, satisfying add(I_f ) cov(I_f ) non(I_f ) cof(I_f ) for every fast increasing f. One technical tool we use is the preservation theory of Judah-Shelah [JS90] and Brendle [Br91]. We generalize this theory so that we can cover the example of preservation presented in [KO14] which is fundamental in this thesis. We also provide original proofs of two Claims from [O08], about the additivity and cofinality of Yorioka ideals, whose proofs have not been published anywhere.