The splitting number can be smaller than the matrix chaos number

Tom 171 / 2002

Heike Mildenberger, Saharon Shelah Fundamenta Mathematicae 171 (2002), 167-176 MSC: 03E15, 03E17, 03E35. DOI: 10.4064/fm171-2-4


Let $\chi $ be the minimum cardinality of a subset of ${}^\omega 2$ that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that ${\frak s} < \chi $ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an $\aleph _2$-iteration of some proper forcing with adding $\aleph _1$ random reals. The second kind of models is obtained by adding $\delta $ random reals to a model of $ {\rm MA}_{<\kappa }$ for some $\delta \in [\aleph _1,\kappa )$. It was a conjecture of Blass that ${\frak s}=\aleph _1 < \chi = \kappa $ holds in such a model. For the analysis of the second model we again use the creature forcing from the first model.


  • Heike MildenbergerInstitut für formale Logik
    Universität Wien
    Währinger Str. 25
    A-1090 Vienna, Austria
  • Saharon ShelahInstitute of Mathematics
    The Hebrew University of Jerusalem
    Givat Ram
    91904 Jerusalem, Israel

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