IV.1 Let us recapitulate. We introduced, in section II.1, the notion of representability, in order to be able to deal formally with those arithmetical concepts, such as that of oddness, that cannot be dealt with explicitly in the formal language. In the following section, we wondered what predicates were representable, seeking a satisfactory understanding of the class of representable predicates; and the considerations raised there, especially Cantor’s argument, forced us to recognize that there are difficulties in the way of this understanding. In the succeeding sections we travelled down what may have seemed a side-track by considering the theory of ∑-forms; it is now time to see that this was no side-track.