Forcing with Urelements

Bokai Yao

Published 2022-12-27Version 1

ZFCU$_{\rm R}$ is ZFC (with the Replacement Scheme) modified to allow a class of urelements. I first isolate a hierarchy of axioms based on ZFCU$_{\rm R}$ and argue that the Collection Principle should be included as an axiom in order to obtain a more robust set theory with urelements. I then turn to forcing over countable transitive models of ZFCU$_{\rm R}$. A new definition of $\mathbb{P}$-names is given. The resulting forcing relation is full just in case the Collection Principle holds in the ground model. While forcing preserves ZFCU$_{\rm R}$ and many axioms in the hierarchy, it can also destroy the DC$_{\omega_1}$-scheme and recover the Collection Principle. The ground model definability fails when the ground model contains a proper class of urelements.