Embedding orders into the cardinals with $\mathsf {DC}_{\kappa} $

Asaf Karagila¹

¹ Einstein Institute of Mathematics Edmond J. Safra Campus The Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel

Fundamenta Mathematicae, Tome 226 (2014) no. 2, pp. 143-156.

Résumé

Jech proved that every partially ordered set can be embedded into the cardinals of some model of $\mathsf {ZF}$. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $\mathsf {ZF}+\mathsf {DC}_{\kappa }$ for any regular $\kappa $. We use this theorem to show that for all $\kappa $, the assumption of $\mathsf {DC}_\kappa $ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model. We use this extension to give a large-cardinals-free proof of independence of the weak choice principle known as $\mathsf {WISC}$ from $\mathsf {DC}_\kappa $.

Détail

DOI : 10.4064/fm226-2-4

Keywords: jech proved every partially ordered set embedded cardinals model mathsf extend result every partially ordered set embedded cardinals model mathsf mathsf kappa regular kappa theorem kappa assumption mathsf kappa does entail there decreasing chains cardinals extend result embed cardinals proper class which definable ground model extension large cardinals free proof independence weak choice principle known mathsf wisc mathsf kappa