 The survey of PM will proceed through the remainder of Volume II and through Volume III, where the theories of rational and real numbers are developed. The contrast intended here is not with Frege’s theories of rational and real numbers, which are present in Grundgesetze but are not seen as a natural extension of the theory of natural numbers. Instead the contemporary account of natural numbers and real numbers is seen as an elementary extension of the axiomatic Zermelo-Frankel set theory. A contemporary textbook in axiomatic set theory, such as Enderton (1977) or Suppes (1960), shows how to construct rational numbers (and negative integers) as pairs of natural numbers, thus 3/4 is constructed as the pair with the operations of addition and multiplication defined as operations on pairs; thus \(1/2 + 1/3 = 10/12 = 5/6\). These positive rational numbers are extended to the whole set by adding negative integers, and then real numbers are defined as Dedekind cuts in the rational numbers, i.e., the set of partitions of sets of rational numbers. The arithmetic of real numbers is then defined for these constructions, and so with sets of real numbers the whole of analysis can be reduced to arithmetic. PM, however, avoids this “arithmetization” of analysis, but instead defines rational, real and in fact a huge class of “relation numbers” as sets of isomorphic sets of relations. Russell says later that he regrets that the theory of relation numbers was not picked up by later set theorists, even though this was some of his most original work in PM. The brief summary of these later topics that we include below, can therefore be seen as a summary of the interesting consequences of taking a different route to the definition of natural numbers based on a logic of relations and properties, rather than the set theory of contemporary foundations of mathematics. This entry is thus aimed at an explication of the unusual order of presentation of these results, in comparison with both Frege and contemporary set theory, and to illustrate these aspects of the theory of relations that are not investigated by contemporary researchers.