The splitting number can be smaller than the matrix chaos number

Heike Mildenberger; Saharon Shelah

Fundamenta Mathematicae (2002)

  • Volume: 171, Issue: 2, page 167-176
  • ISSN: 0016-2736

Abstract

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Let χ be the minimum cardinality of a subset of ω 2 that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ 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 ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of M A < κ for some δ ∈ [ℵ₁,κ). It was a conjecture of Blass that = ℵ₁ < χ = κ holds in such a model. For the analysis of the second model we again use the creature forcing from the first model.

How to cite

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Heike Mildenberger, and Saharon Shelah. "The splitting number can be smaller than the matrix chaos number." Fundamenta Mathematicae 171.2 (2002): 167-176. <http://eudml.org/doc/283065>.

@article{HeikeMildenberger2002,
abstract = {Let χ be the minimum cardinality of a subset of $^ω 2$ that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ 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 ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of $MA_\{<κ\}$ for some δ ∈ [ℵ₁,κ). It was a conjecture of Blass that = ℵ₁ < χ = κ holds in such a model. For the analysis of the second model we again use the creature forcing from the first model.},
author = {Heike Mildenberger, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {cardinal invariants; chaos number; Toeplitz matrix; consistency; random reals; forcing},
language = {eng},
number = {2},
pages = {167-176},
title = {The splitting number can be smaller than the matrix chaos number},
url = {http://eudml.org/doc/283065},
volume = {171},
year = {2002},
}

TY - JOUR
AU - Heike Mildenberger
AU - Saharon Shelah
TI - The splitting number can be smaller than the matrix chaos number
JO - Fundamenta Mathematicae
PY - 2002
VL - 171
IS - 2
SP - 167
EP - 176
AB - Let χ be the minimum cardinality of a subset of $^ω 2$ that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ 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 ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of $MA_{<κ}$ for some δ ∈ [ℵ₁,κ). It was a conjecture of Blass that = ℵ₁ < χ = κ holds in such a model. For the analysis of the second model we again use the creature forcing from the first model.
LA - eng
KW - cardinal invariants; chaos number; Toeplitz matrix; consistency; random reals; forcing
UR - http://eudml.org/doc/283065
ER -

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