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2 results

  • A Forcing Axiom Deciding the Generalized Souslin Hypothesis

  • Part of
  • Chris Lambie-Hanson, Assaf Rinot
  • Journal:
    Canadian Journal of Mathematics / Volume 71 / Issue 2 / April 2019
    Published online by Cambridge University Press:
    07 January 2019, pp. 437-470
    Print publication:
    April 2019
    • Article
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  • We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\unicode[STIX]{x1D706}$, if $\unicode[STIX]{x1D706}^{++}$ is not a Mahlo cardinal in Gödel’s constructible universe, then $2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$ entails the existence of a $\unicode[STIX]{x1D706}^{+}$-complete $\unicode[STIX]{x1D706}^{++}$-Souslin tree.

  • Aronszajn lines and the club filter

  • Justin Tatch Moore
  • Journal:
    The Journal of Symbolic Logic / Volume 73 / Issue 3 / September 2008
    Published online by Cambridge University Press:
    12 March 2014, pp. 1029-1035
    Print publication:
    September 2008

  • The purpose of this note is to demonstrate that a weak form of club guessing on ω1 implies the existence of an Aronszajn line with no Countryman suborders. An immediate consequence is that the existence of a five element basis for the uncountable linear orders does not follow from the forcing axiom for ω-proper forcings.