Bokai Yao

Forcing with Urelements

I will begin by isolating a hierarchy of axioms based on ZFCU_R, which is ZFC set theory (with Replacement) modified to allow a class of urelements. For example, the Collection Principle is equivalent to the Reflection Principle over ZFCU_R, while it is folklore that neither of them is provable in ZFCU_R.

I then turn to forcing over countable transitive models of ZFU_R. A forcing relation is full just in case whenever a forcing condition p forces an existential statement, p also forces some instance of that statement. According to the existing approach, forcing relations are almost never full when there are urelements. I introduce a new forcing machinery to address this problem. I show that over ZFCU_R, the principle that every new forcing relation is full is equivalent to the Collection Principle. Furthermore, I show how forcing is able to preserve, destroy and resurrect the axioms in the hierarchy I introduced. In particular, the Reflection Principle is “necessarily forceble” in certain models of ZFCU_R. In the end, I will consider how the ground model definability can fail when the ground model contains a proper class of urelements.

The zoom link for this talk is https://bristol-ac-uk.zoom.us/j/93185058773.