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On partial orderings having precalibre-$\aleph _1$ and fragments of Martin's axiom

Volume 232 / 2016

Joan Bagaria, Saharon Shelah Fundamenta Mathematicae 232 (2016), 181-197 MSC: Primary 03Exx; Secondary 03E50, 03E57. DOI: 10.4064/fm232-2-6

Abstract

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph _1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $\sigma $-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for $\sigma $-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprāns and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph _1$ property of a partial ordering while preserving its ccc-ness.

Authors

  • Joan BagariaICREA (Institució Catalana de Recerca
    i Estudis Avançats)
    and
    Departament de Lògica, Història
    i Filosofia de la Ciència
    Universitat de Barcelona
    Montalegre 6
    08001 Barcelona, Catalonia, Spain
    e-mail
    e-mail
  • Saharon ShelahEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Edmond J. Safra Campus
    Givat Ram, Jerusalem 91904, Israel
    e-mail

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