Boolean-Valued Models of Set Theory with Urelements

We explore Boolean-valued models of set theory with a class of urelements. In an existing construction, which we call U-B, every urelement is its own B-name. We prove the fundamental theorem of U-B in the context of ZFU(R) (i.e., ZF with urelements formulated with Replacement). In particular, U-B is shown to preserve Replacement and hence ZFUR. Moreover, U-B can both destroy axioms, such as the DC omega 1-scheme, and recover axioms, such as the Collection Principle. One drawback of U-B is that it does not permit mixing names, resulting in a lack of fullness. To address this, we introduce a new construction, U-B, which is closed under mixtures. We prove that there is an elementary embedding from U-B to U-B. Over ZFU(R) with the Axiom of Choice, U-B is full for every complete Boolean algebra B just in case the Collection Principle holds.