The Ordering Principle and Higher Dependent Choice

Abstract

We provide, for any regular uncountable cardinal $\hat{I}º$, a new argument for Pincus' result on the consistency of $\mathrm{ZF}$ with the higher dependent choice principle ${\mathrm{DC}}_{<\hat{I}º}$ and the ordering principle in the presence of a failure of the axiom of choice. We also generalise his methods and obtain these consistency results in a larger class of models.

13 pages