Embedding orders into the cardinals with $\mathsf {DC}_{\kappa} $
Tom 226 / 2014
Fundamenta Mathematicae 226 (2014), 143-156
MSC: Primary 03E25; Secondary 03E35.
DOI: 10.4064/fm226-2-4
Streszczenie
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 $.
