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Free sequences in \({\mathscr {P}}\left( \omega \right) /\text {fin}\)

  • David ChodounskýEmail author
  • Vera Fischer
  • Jan Grebík
Article
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Abstract

We investigate maximal free sequences in the Boolean algebra \({\mathscr {P}}\left( \omega \right) {/}\text {fin}\), as defined by Monk (Comment Math Univ Carol 52(4):593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted \({\mathfrak {f}}\). Answering a question of Monk, we demonstrate the consistency of \(\omega _1 = {\mathfrak {i}} = {\mathfrak {f}} < {\mathfrak {u}} = \omega _2\). In fact, this consistency is demonstrated in the model of Shelah for \({\mathfrak {i}}< {\mathfrak {u}}\) (Shelah in Arch Math Log 31(6):433–443, 1992). Our paper provides a streamlined and mostly self contained presentation of this construction.

Keywords

Maximal free sequence Dense independent system Party forcing 

Mathematics Subject Classification

03E17 03E35 06E05 

Notes

Acknowledgements

The authors would like to thank Osvaldo Guzmán for numerous suggestions substantially improving the paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics of the Czech Academy of SciencesPrague 1Czech Republic
  2. 2.Kurt Gödel Research CenterUniversity of ViennaViennaAustria

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