 Admittedly the system of PM is an indirect and cumbersome system to develop if the theory of Arithmetic were the only goal in mind. Firstly, however, the system of the ramified theory of types is independently interesting for the foundations of logic that it provides. After ∗20 the theory of classes, and development of general notions of arithmetic for relations which follows, does present the arithmetic of the natural numbers as a special case which can be generalized to the arithmetic of ordinal and cardinal numbers all in a logic with a simple hierarcy of types. The survey below of what follows in Volume II and Volume III shows the particular way of developing the theory of rational and then real numbers that PM follows. The results in set theory will seem primitive, as the results are dated to around 1908, at just the point when axiomatic set theory began its extraordinary development. Whitehead and Russell were not active contributors to set theory and so PM should not be studied for later technical results that may have been anticipated here. Russell summarized results in PM from the current state of the study of infinite cardinals and ordinals in a paper he give in Paris in a paper called “On the Axioms of the Infinite and Transfinite” (Russell 1911). There is, however, one result concerning two notions of infinity that appears to originate with PM.