 Dedekind proposed that the real numbers could be uniquely correlated with cuts in the rationals, where a cut was determined by a pair of sets \((L, R)\) with the following properties: \(L\) and \(R\) are sets of rationals. \(L\) and \(R\) are both nonempty and every element of \(L\) is less than every element of \(R\) (so the two sets are disjoint). \(L\) has no greatest element. The union of \(L\) and \(R\) contains all rationals.