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Abstract

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 $.