Philosophy of Mathematics seminar




Xinhe Wu, Bristol
BooleanValued Models of Set Theory with Urelements
We study Booleanvalued models of set theory with a proper class of urelements. We prove the fundamental theorem for Booleanvalued models with urelements concerning axiom preservation over ZFCU_{R}. We show that certain axioms such as the DC_{ω1} scheme are preserved only by certain complete Boolean algebras. We then turn to the property of fullness. Since the standard Booleanvalued 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 ZFCU_{R}, the Axiom of Collection is equivalent to a principle concerning the fullness of the new construction.
This is joint work with Bokai Yao.

