We have the ordered field of [[Rational Number|rational numbers]], but they aren’t complete there are holes everywhere, and to get to the [[Real numbers|set of reals]] $\mathbb{R}$ we must fill in these gaps. There are several ways to do this ([[@cummingsRealAnalysisLongform2021]]): - There is also a non-standard approach using so-called ultrafilters, another using hyperrationals, and several more beyond that. - Cantor used Cauchy sequences of rationals: If you demand that each Cauchy sequence converges to something, and pairs of sequences whose difference is Cauchy must converge to the same thing, then by adding these limits into your set Q (by identifying them with an equivalence class of Cauchy sequences converging to them), you in effect complete Q, giving R. - Another common method uses *Dedekind cuts.* **Definitions** 1. [[The set of real numbers is the set of Dedekind cuts]]. 2. [[Cauchy's construction of real numbers]].