 A bolder move (given a theory of the rationals but no prior theory of the reals) is to define the real numbers as cuts. Notice that this requires us to have not only a theory of the rational numbers (not difficult to develop) but also a theory of sets of rational numbers: if we are to understand a real number to be identified with a cut in the rational numbers, where a cut is a pair of sets of rational numbers, we do need to understand what a set of rational numbers is. If we are to demonstrate the existence of particular real numbers, we need to have some idea what sets of rational numbers there are.