Embedding Orders Into Cardinals With $DC_\kappa$

Abstract

Jech proved that every partially ordered set can be embedded into the cardinals of some model of $ZF$. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $ZF+DC_{<\kappa}$ for any regular $\kappa$. We use this theorem to show that for all $\kappa$, the assumption of $DC_\kappa$ 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 $WISC$.