Matrix iterations and Cichon’s diagram

Abstract

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon's diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in [4], we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon's diagram.

Creature forcing and five cardinal characteristics in Cichoń’s diagram

“…13 The reason for the logarithm, and the use of the halving parameters, will become clear only in Sect. 5.2.…”

On cardinal characteristics of Yorioka ideals

“…Concerning the problem of separating the four cardinal invariants associated with an ideal, the consistency of is known from . Question Is there a model of satisfying …”

“…All the results of this subsection are versions of the contents of [, § 4] in the context of generalized Polish relational system. Though the proofs are similar, we still present them for completeness.…”

“…Concerning problems of this nature, the consistency of with is a consequence of [, Theorem 17]. Quite recently, Goldstern, Kellner and Shelah showed the consistency, modulo strongly compact cardinals, of Cichoń's diagram separated into 10 different values, in particular, the four cardinal invariants associated with the meager ideal on are pairwise different is consistent .…”

“…Later on, this method was improved by Brendle and Fischer when they proved the consistency of and with large continuum, the latter assuming the existence of a measurable cardinal in the ground model . The second author induced known preservation results in such type of iterations to construct models where several cardinals in Cichoń's diagram are pairwise different . Just a while ago, Dow and Shelah used this technique to prove that the splitting number is consistently singular .…”