Philosophy of Mathematics Seminar


Philosophy of Mathematics seminar


Xinhe Wu, Bristol

Boolean-Valued Models of Set Theory with Urelements

We study Boolean-valued models of set theory with a proper class of urelements. We prove the fundamental theorem for Boolean-valued models with urelements concerning axiom preservation over ZFCUR. We show that certain axioms such as the DCω1 scheme are preserved only by certain complete Boolean algebras. We then turn to the property of fullness. Since the standard Boolean-valued models with urelements are almost never full, we provide a different construction. The standard construction is shown to be an elementary substructure of the new construction. Finally, we prove that over ZFCUR, the Axiom of Collection is equivalent to a principle concerning the fullness of the new construction.

This is joint work with Bokai Yao.