Locally $\Sigma _{1}$-definable well-orders of ${\rm H}(\kappa ^+)$
Volume 226 / 2014
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.
