The splitting number can be smaller than the matrix chaos number (Q2778631)

For an \(\omega\times\omega\) matrix \(A=(a_{i,j})_{i,j<\omega}\) of reals and \(f\in{}^\omega\mathbb R\), \(A\lim f=\lim_{i\to\infty}\sum_{j=0}^\infty a_{i,j}f(j)\). The matrices \(A\) for which the operator \(A\lim\) extends the ordinary limit operator are exactly the so-called Toeplitz matrices. The cardinal number considered in the paper is the chaos number \(\chi\), the minimal size of a family \(\mathcal F\subseteq{}^\omega 2\) such that for every Toeplitz matrix \(A\) there is \(f\in\mathcal F\) such that \(A\lim f\) does not exist. This cardinal invariant was introduced by P. Vojtáš who proved that \(\mathfrak s\leq\chi\leq\mathfrak b\cdot\mathfrak s\) where \(\mathfrak s\) and \(\mathfrak b\) are the so-called splitting number and the unbounding number, respectively. The first of the authors proved [\textit{H. Mildenberger}, Fundam. Math. 165, 175-189 (2000; Zbl 0959.03036)] that \(\chi<\mathfrak b\cdot\mathfrak s\) is consistent relative to ZFC. In the paper under review, the authors prove the consistency of the inequality \(\mathfrak s<\chi\). In fact, they give two different models for this inequality. The first is obtained by adding random reals to a model obtained by a countable support iteration of length \(\omega_2\) of the creature forcing, which is a special case of a more extensive framework [\textit{S. Shelah} and \textit{A. Rosłanowski}, Norms on possibilities. I: Forcing with trees and creatures, Mem. Am. Math. Soc. 671 (1999; Zbl 0940.03059)]. The second model is obtained by adding random reals to a model of MA\(_{<\kappa}\). This model was previously conjectured by A. Blass.