 Like  Gottlob Frege,  Russell’s basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity. Thus the number 1 is to be identified with the class of all unit classes, the number 2 with the class of all two-membered classes, and so on. Statements such as “There are at least two books” would be recast as statements such as “There is a book, x, and there is a book, y, and x is not identical to y.” Statements such as “There are exactly two books” would be recast as “There is a book, x, and there is a book, y, and x is not identical to y, and if there is a book, z, then z is identical to either x or y.” It follows that number-theoretic operations may then be explained in terms of set-theoretic operations such as intersection, union, and difference. In Principia Mathematica, Whitehead and Russell were able to provide many detailed derivations of major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. They were also able to develop a sophisticated theory of logical relations and a unique method of founding the real numbers. Even so, the issue of whether set theory itself can be said to have been successfully reduced to logic remained controversial. A fourth volume on geometry was planned but never completed.