# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## The splitting number can be smaller than the matrix chaos number

### Tom 171 / 2002

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

#### Streszczenie

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.

#### Autorzy

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

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek