Let us emphasize that it is not claimed that, e.g., real numbers are Dedekind cuts of rationals, as they could also be defined using Cauchy sequences, or in other different ways. What is important, from a foundational point of view, is that the set-theoretic version of \(\mathbb{R}\), together with the usual algebraic operations, satisfies the categorical axioms that the real numbers satisfy, namely those of a complete ordered field. The metaphysical question of what the real numbers really are is irrelevant here.