Ken’s immediate reason for introducing the pred operator at that point is to be able to state Theorem I.6.3, which is fundamental to any understanding of well-orders: if $\langle X,\le\rangle$ and $\langle Y,\preceq\rangle$ are well-orders, then either they are isomorphic, or one is isomorphic to a proper initial segment of the other. In the very next section he goes on to construct the ordinals $-$ transitive sets $\alpha$ such that $\langle \alpha,\in\rangle$ is a well-order $-$ and shows that every well-order is isomorphic to one of them. This improves on Theorem I.6.3, because if $\alpha$ and $\beta$ are ordinals, then either they are equal (not just isomorphic), or one of them is a proper initial segment of the other (not just isomorphic to one). Moreover, each ordinal is a (necessarily proper!) initial segment of the proper class ON of ordinals. Thus, the notion of initial segment is intimately bound up with the concept of a well-order, which in turn is fundamental to set theory.