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Construction with opposition: cardinal invariants and games

  • Jörg Brendle
  • Michael Hrušák
  • Víctor Torres-PérezEmail author
Open Access
Article
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Abstract

We consider several game versions of the cardinal invariants \({\mathfrak {t}}\), \({\mathfrak {u}}\) and \({\mathfrak {a}}\). We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \({\mathfrak {t}}<{\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}<{\mathfrak {u}}_{Builder}\) are both relatively consistent with ZFC, where \({\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}_{Builder}\) are the principal game versions of \({\mathfrak {t}}\) and \({\mathfrak {u}}\), respectively. The corresponding question for \({\mathfrak {a}}\) remains open.

Keywords

Cardinal invariants of the continuum Transfinite games Parametrized diamond principles 

Mathematics Subject Classification

03E05 03E15 03E17 

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF).

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Graduate School of System InformaticsKobe UniversityKobeJapan
  2. 2.Centro de Ciencias MatemáticasUNAMMoreliaMexico
  3. 3.Institute of Discrete Mathematics and GeometryTU WienViennaAustria

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