   In model theory, one can take values in any complete BA rather than the two-element BA. This Boolean-valued model theory was developed around 1950–1970, but has not been worked on much since. But a special case, Boolean-valued models for set theory, is very much at the forefront of current research in set theory. It actually forms an equivalent way of looking at the forcing construction of Cohen, and has some technical advantages and disadvantages. Philosophically it seems more satisfactory than the forcing concept. We describe this set theory case here; it will then become evident why only complete BAs are considered. Let B be a complete BA. First we define the Boolean valued universe \(V(B)\). The ordinary set-theoretic universe can be identified with \(V\)(2), where 2 is the 2-element BA. The definition is by transfinite recursion, where \(\alpha , \beta\) are ordinals and \(\lambda\) is a limit ordinal: