Dedekind cut
For convenience we may take the lower set formula_1 as the representative of any given Dedekind cut formula_2, since formula_3 completely determines formula_4. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number formula_5 is any subset of the set formula_6 of rational numbers that fulfills the following conditions: