March 13
Bokai Yao,
Peking University
Reflection Principles in ZFU
We study first-order reflection principles in ZF set theory with urelements (ZFU). The Reflection Principle (RP) asserts that every formula is absolute between the universe and some transitive set, while the Partial Reflection Principle (RP-) states that every true assertion is true in some transitive set. It is known that RP-, RP, and the Collection Principle are equivalent over ZFU with the Axiom of Choice, yet none of these principles is provable. We separate these three principles in ZFU: Collection and RP- are independent of one another, and Collection together with RP- does not imply RP. We also show that RP and Collection are equivalent assuming either the Tail axiom or Small Violations of Choice. This is joint work with Elliot Glazer.