Simplest possible locally definable well-orders

Peter Holy¹ ; Philipp Lücke¹

¹ Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn Endenicher Allee 60 53115 Bonn, Germany

Fundamenta Mathematicae, Tome 236 (2017) no. 2, pp. 101-139

Résumé

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.

Détail

DOI : 10.4064/fm281-7-2016

Keywords: study provable consequences existence well order kappa definable sigma formula structure langle kappa rangle where kappa uncountable regular cardinal accomplished constructing partial orders force existence well orders while preserving many structural features ground model these constructions existence well order omega definable langle omega rangle sigma formula parameter omega consistent failure gch omega moreover achieve situation presence measurable cardinal contrast results woodin imply existence well order incompatible existence infinitely many woodin cardinals measurable cardinal above