PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Simplest possible locally definable well-orders

Volume 236 / 2017

Peter Holy, Philipp Lücke Fundamenta Mathematicae 236 (2017), 101-139 MSC: 03E35, 03E47, 03E50. DOI: 10.4064/fm281-7-2016 Published online: 2 November 2016

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.

Authors

  • Peter HolyMathematisches Institut
    Rheinische Friedrich-Wilhelms-Universität Bonn
    Endenicher Allee 60
    53115 Bonn, Germany
    e-mail
  • Philipp LückeMathematisches Institut
    Rheinische Friedrich-Wilhelms-Universität Bonn
    Endenicher Allee 60
    53115 Bonn, Germany
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image