Boolean-valued model

Summary

A Boolean-valued model is like a first-order structure, but each statement gets a truth value in a complete Boolean algebra. In an ordinary structure the only values are "true" and "false". When the Boolean algebra has two elements, it is essentially an ordinary structure. Boolean-valued models of set theory, written $V^B$, give a way to present forcing that needs no countable transitive model. They were developed by Scott, Solovay and Vopěnka.