Some combinatorial principles defined in terms of elementary submodels

Volume 181 / 2004

Sakaé Fuchino, Stefan Geschke Fundamenta Mathematicae 181 (2004), 233-255 MSC: Primary 03E35; Secondary 03E05, 03E17, 03E65. DOI: 10.4064/fm181-3-3


We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese–Nation property of ${\mathcal P}(\omega )$ studied in [6] already follow from SEP. We show that it is consistent that SEP holds while ${\mathcal P}(\omega )$ fails to have the $(\aleph _1,\aleph _0)$-ideal property introduced in [2]. This answers a question addressed independently by Fuchino and by Kunen. We also consider some natural variants of SEP and show that certain changes in the definition of SEP do not lead to a different principle, answering a question of Blass.


  • Sakaé FuchinoDepartment of Natural Science and Mathematics
    College of Engineering
    Chubu University
    Kasugai, Aichi 487-8501, Japan
  • Stefan GeschkeII. Mathematisches Institut
    Freie Universität Berlin
    Arnimallee 3
    14195 Berlin, Germany

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