I came up with a different definition that is a kind of inverse of Dedekind cuts. It is the idea that a real number is the set of all rational intervals that contain it. Since this is circular, there are properties that I came up with which say when a set of rational intervals qualifies to be called a real number in my setup. I have an unreviewed paper which creates a version that is a bridge between numerical analysis and the theoretical definition of a real number. Another unreviewed paper shows the equivalence between my definition and Dedekind cuts. You can read both at [1].

There is a long tradition of using intervals for dealing with real numbers. It is often used by constructivists and can be thought of viewing a real number as a measurement.

1: https://github.com/jostylr/Reals-as-Oracles