 The paperback abridged edition of Principia Mathematica to ∗56 only goes this far, so the remaining definitions have only been available to those with access to the full three volumes of PM. Russell did not make the decision to end the 1962 abridged version at this point, but the choice is understandable. It is here that contemporary set theory begins to look even more different from PM. Set theory follows Norbert Wiener (1914) by representing relations as sets of ordered pairs, which themselves defined as sets. (Wiener’s proposal of \( \langle x, y \rangle =_\mathit{df} \{ \{ \{x \}, \emptyset \}, \{ \{y \} \} \} \) has generally been replaced by Kuratowski’s simpler \(\{ \{x \}, \{ x,y \} \}\)) . The remainder of PM examines the structure of relations that lead to the mathematics of natural and real numbers, and the portion of the theory of transfinite sets that can be carried out in the theory of types. This looks very different from the development of these notions in axiomatic set theory. 