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

Peter Holy ¹   ; Philipp Lücke ²

¹ School of Mathematics University of Bristol University Walk Bristol BS8 1TW, United Kingdom
² Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn Endenicher Allee 60 53115 Bonn, Germany

Fundamenta Mathematicae, Tome 226 (2014) no. 3, pp. 221-236

Résumé

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.

Détail

DOI : 10.4064/fm226-3-2

Keywords: given uncountable cardinal kappa kappa kappa kappa kappa regular there forcing preserves cofinalities equal kappa forces existence well order kappa definable langle kappa rangle sigma formula parameters shows contrast kappa omega existence locally definable well order kappa low complexity consistent failures gch kappa forcing mentioned above introduces bernstein subset kappa kappa definable langle kappa rangle delta formula parameters