Embedding Orders Into Cardinals With $DC_\kappa$

TL;DR

This paper extends Jech's embedding theorem to models with weaker choice axioms, demonstrating that partially ordered sets can be embedded into cardinals under $ZF+DC_{<\kappa}$ and exploring implications for decreasing chains and large cardinals.

Contribution

It generalizes embedding results to models with $ZF+DC_{<\kappa}$ and provides a large cardinals-free proof of the independence of $WISC$.

Findings

Every poset embeds into cardinals of models with $ZF+DC_{<\kappa}$.

$DC_\kappa$ does not prevent decreasing chains of cardinals.

Embedding of definable proper classes into cardinals.

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 $WISC$.