 Although students of philosophy usually read no further than ∗20 in PM, this is in fact the point where the “construction” of mathematics really begins. ∗21 presents the “General Theory of Relations” (the theory of relations in extension; in contemporary logic these are treated as sets of ordered pairs, following Wiener). \(\hat{x} \hat{y} \psi(x, y)\) is the relation between \(x\) and \(y\) which obtains when \(\psi(x, y)\) is true. In modern notation we represent this as as the set of ordered pairs \(\{\langle x, y \rangle \mid \psi(x, y)\}\), which is read: the set of ordered pairs \(\langle x, y \rangle\) which are such that \(x\) bears the relation \(\psi\) to \(y\). 