Abstract: The construction of V^(B) is relativized to transitive models of ZFC and shown to give rise to actual models of ZFC, in which various set-theoretic assertions are falsified. Boolean-valued set theory is thereby transformed into a valuable model-theoretic tool. The notion of a generic ultrafilter in a Boolean-valued model is introduced in this chapter and shown to play a key role in the discussion.

Keywords: transitive model, Boolean-valued set theory, generic ultrafilter, ZFC,