Dedekind cut
As an example of a Dedekind cut representing an irrational number, we may take the positive square root of 2. This can be defined by the set formula_49. It can be seen from the definitions above that formula_3 is a real number, and that formula_51. However, neither claim is immediate. Showing that formula_1 is real requires showing that formula_3 has no greatest element, i.e. that for any positive rational formula_54 with formula_55, there is a rational formula_56 with formula_57 and formula_58 The choice formula_59 works. Then formula_60 but to show equality requires showing that if formula_61 is any rational number with formula_62, then there is positive formula_54 in formula_3 with formula_65.