A model of the real number set (Q2730535)

Let \(Q\) be the set of rational numbers and for any set \(A\subset Q\) let \(A'=Q\setminus A\), \(A<B\Longleftrightarrow (\forall a\in A)(\forall b\in B) a<b\). The set \(A\) is called Dedekind cutset if \(A'<A\) and \(\overline\exists \max A'\). The set of all Dedekind cutsets is denoted by \({\mathcal R}\). The author defines the operations of addition, multiplication and proves that \({\mathcal R}\) with these operation is a field and it is a model of the real number set.