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Locally $\Sigma _{1}$-definable well-orders of ${\rm H}(\kappa ^+)$

Volume 226 / 2014

Peter Holy, Philipp Lücke Fundamenta Mathematicae 226 (2014), 221-236 MSC: 03E35, 03E47. DOI: 10.4064/fm226-3-2

Abstract

Given an uncountable cardinal $\kappa$ with $\kappa=\kappa^{{<}\kappa}$ and $2^\kappa$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^\kappa$ and forces the existence of a well-order of ${\rm H}(\kappa^+)$ that is definable over $\langle{\rm H}(\kappa^+),\in\rangle$ by a $\Sigma_1$-formula with parameters. This shows that, in contrast to the case “$\kappa=\omega$”, the existence of a locally definable well-order of ${\rm H}(\kappa^+)$ of low complexity is consistent with failures of the ${\rm GCH}$ at $\kappa$. We also show that the forcing mentioned above introduces a Bernstein subset of ${}^\kappa\kappa$ that is definable over $\langle{\rm H}(\kappa^+),\in\rangle$ by a $\Delta_1$-formula with parameters.

Authors

  • Peter HolySchool of Mathematics
    University of Bristol
    University Walk
    Bristol BS8 1TW, United Kingdom
    e-mail
  • Philipp LückeMathematisches Institut
    Rheinische Friedrich-Wilhelms-Universität Bonn
    Endenicher Allee 60
    53115 Bonn, Germany
    e-mail

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