  Members of V(B) are called B-valued sets. It is now easily seen that a B-valued set is precisely a B-valued function with domain a set of B-valued sets. Now let L be the first-order language of set theory and let L(B) be the language obtained by adding to L a name for each element of V(B) (we shall use the same symbol for the element and its name). One can now construct a mapping [·](B) of the (sentences of the) language L(B) into B: for each sentence σ of L(B), the element [σ](B) of B is the “Boolean truth value” of σ in V(B). This mapping [·](B) is defined so as to send all the theorems of Zermelo-Fraenkel set theory to the top element 1 of B, i.e., to “truth”; accordingly, V(B) may be thought of as a Boolean-valued model of set theory. In general, if [σ](B) = 1, we say that σ is valid in V(B), and write V(B)  ⊨ σ.