 The real numbers \(\Theta\) are defined as “the series of segments of the series of ratios”, or the set of Dedekind cuts of the sequence of rational numbers, in their standard ordering. Technically \(\Theta\) is a relation in extension. The individuals that are related in the ratios, and thus at the basis of the series of rational numbers must be infinite in number for the PM version of the real numbers to have the structure we expect. In the introductory material to the section they point out that while the construction of the reals thus requires the Axiom of Infinity, they add it explicitly to the theorems where needed and try as much as possible to derive the results upon which it does not rely without making that assumption.