The splitting number can be smaller than the matrix chaos number

Heike Mildenberger¹ ; Saharon Shelah²

¹ Institut für formale Logik Universität Wien Währinger Str. 25 A-1090 Vienna, Austria
² Institute of Mathematics The Hebrew University of Jerusalem Givat Ram 91904 Jerusalem, Israel

Fundamenta Mathematicae, Tome 171 (2002) no. 2, pp. 167-176.

Résumé

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.

Détail

DOI : 10.4064/fm171-2-4

Keywords: chi minimum cardinality subset omega cannot made convergent multiplication single toeplitz matrix application creature forcing frak chi consistent answer question vojt kinds models strict inequality first combination aleph iteration proper forcing adding aleph random reals second kind models obtained adding delta random reals model kappa delta aleph kappa conjecture blass frak aleph chi kappa holds model analysis second model again creature forcing first model