Boolean-Valued Models of Set Theory with Urelements

Abstract

We explore Boolean-valued models of set theory with a class of urelements. In an existing construction, which we call $U^{\mathbb{B}}$, every urelement is its own $\mathbb{B}$-name. We prove the fundamental theorem of $U^{\mathbb{B}}$ in the context of ${\mathrm{ZFU}}_{\mathrm{R}}$ (i.e., ZF with urelements formulated with Replacement). In particular, $U^{\mathbb{B}}$ is shown to preserve Replacement and hence ${\mathrm{ZFU}}_{\mathrm{R}}$. Moreover, $U^{\mathbb{B}}$ can both destroy axioms, such as the DC${}_{{\mathit{\omega }}_1}$-scheme, and recover axioms, such as the Collection Principle. One drawback of $U^{\mathbb{B}}$ is that it does not permit mixing names, resulting in a lack of fullness. To address this, we introduce a new construction, $\overline{U^{\mathbb{B}}}$, which is closed under mixtures. We prove that there is an elementary embedding from $U^{\mathbb{B}}$ to $\overline{U^{\mathbb{B}}}$. Over ${\mathrm{ZFU}}_{\mathrm{R}}$ with the Axiom of Choice, $\overline{U^{\mathbb{B}}}$ is full for every complete Boolean algebra $\mathbb{B}$ just in case the Collection Principle holds.