 The writing of this preface delayed the publication of the second volume of PM, as Whitehead and Russell struggled over the complications it raised. The difficulties arise from the typical ambiguity of terms and formulas of the theory of types. Every constant, such as those for the numbers \(0,1, \ldots, \aleph_0\) will have a definition relative to each type. Without assuming the Axiom of Infinity for individuals, there is no guarantee that a given constant designates a non-empty class in a given type. The preface introduces the notion of “formal numbers”, which are to be interpreted as belonging to a type that makes them not identical with the \(\emptyset\) for that type. Volume II begins with Part III, “Cardinal Arithmetic”. The notions of cardinal numbers are developed in full generality, extending to infinite cardinals. Consequently the theory of natural numbers, which are called “Inductive Cardinals” in PM, is introduced with a series of definitions of special cases of notions that are first introduced in a general form applying to any numbers or classes. For example, addition of natural numbers, as in the famous proof that \(1 + 1 = 2\) in ∗110·04 is proved  for the special case of the addition of classes that applies to cardinal numbers, ‘\(+_c\)’. The Summary to section A introduces the notion of homogenous cardinals, which are classes of similar classes whose members are all of the same type. It is possible to define similarity between two classes \(\alpha\) and \(\beta\) of distinct types say \(\tau\) and \(\tau '\), and cardinals are classified as descending and ascending as the domain of the relevant similarity relation is of a higher type than the range, and when of a lower type, respectively. The theory of cardinal numbers is straightforward with homogenous cardinals, however the exceptions must be kept in mind, as is evidenced in ∗100.