Simplest possible locally definable well-orders
Volume 236 / 2017
Abstract
We study the provable consequences of the existence of a well-order of ${\rm {H}}(\kappa ^+)$ definable by a $\Sigma _1$-formula over the structure $\langle {\rm {H}}(\kappa ^+),\in \rangle $ in the case where $\kappa $ is an uncountable regular cardinal. This is accomplished by constructing partial orders that force the existence of such well-orders while preserving many structural features of the ground model. We will use these constructions to show that the existence of a well-order of ${\rm {H}}(\omega _2)$ that is definable over $\langle {\rm {H}}(\omega _2),\in \rangle $ by a $\Sigma _1$-formula with parameter $\omega _1$ is consistent with a failure of the ${\rm {GCH}}$ at $\omega _1$. Moreover, we will show that one can achieve this situation also in the presence of a measurable cardinal. In contrast, results of Woodin imply that the existence of such a well-order is incompatible with the existence of infinitely many Woodin cardinals with a measurable cardinal above them all.
