Dedekind's construction replaces an intuitive continuum by an order-theoretic object. A real number is encoded by the rational numbers lying strictly to its left.
Cuts as left halves of rational order
A cut records all rational approximations below a would-be real number.
Definition. A Dedekind cut is a nonempty proper subset \(\alpha\subset\mathbb Q\) such that \(q\in\alpha\) and \(r\lt q\) imply \(r\in\alpha\), and such that \(\alpha\) has no greatest element.
Example. For \(a\in\mathbb Q\), the rational cut is \(\alpha_a=\{q\in\mathbb Q:q\lt a\}\). The irrational cut for \(\sqrt2\) is \(\{q:q\lt 0\}\cup\{q\ge0:q^2\lt 2\}\).
Order and rational embedding
The order of cuts is inclusion; larger cuts contain more rational numbers.
Proposition. The map \(a\mapsto\alpha_a\) embeds \(\mathbb Q\) into the set of cuts, and \(\alpha_a\subset\alpha_b\) if and only if \(a\le b\).
Proof. If \(a\le b\), then every rational below \(a\) is below \(b\). If \(\alpha_a\subset\alpha_b\) and \(b\lt a\), density of \(\mathbb Q\) gives \(q\) with \(b\lt q\lt a\); then \(q\in\alpha_a\) but \(q\notin\alpha_b\), contradiction.
Arithmetic of cuts
Addition and multiplication are first defined on positive cuts, then extended by signs.
Definition. For cuts \(\alpha,\beta\), define \(\alpha+\beta=\{r\in\mathbb Q:r\lt a+b\text{ for some }a\in\alpha,b\in\beta\}\). For positive cuts, define \(\alpha\beta=\{r:r\lt ab\text{ for some }a\in\alpha,b\in\beta,\ a,b>0\}\), with the usual sign extension.
Proposition. With these operations and inclusion order, Dedekind cuts form a complete ordered field.
Proof. For addition, nonemptiness and downward closure are immediate from the definition. If \(r<a+b\), rational density supplies \(r<r'<a+b\), so the sum cut has no greatest element. Associativity and commutativity follow by regrouping rational approximants, and \(\alpha_0\) is the additive identity.
Define \[ -\alpha=\{q\in\mathbb Q:\text{some }r>q \text{ satisfies }-r\notin\alpha\}. \] This strict formulation removes a possible greatest element. Directly from downward closure, \(-\alpha\) is a cut and \(\alpha+(-\alpha)=\alpha_0\).
For positive cuts, products of positive approximants define a cut; rational density again removes boundary points. Distributivity and associativity follow from their rational counterparts. If \(\alpha>\alpha_0\), define \[ \alpha^{-1} =\{q\le0\}\cup \{q>0:\text{some }r>q\text{ has }r^{-1}\notin\alpha\}. \] Approximation from both sides of the boundary proves \(\alpha\alpha^{-1}=\alpha_1\). Sign rules extend multiplication and inverses to all nonzero cuts.
Inclusion is a total order because two lower sets with no greatest element cannot cross: if \(a\in\alpha\setminus\beta\), then every \(b\in\beta\) satisfies \(b<a\), and downward closure gives \(\beta\subset\alpha\). The definitions of sum and positive product preserve inclusion. Hence the cuts form an ordered field. Completeness follows from the union construction below.
Suprema by unions
The crucial advantage of cuts is that bounded families have visibly defined suprema.
Theorem. If \(\mathcal A\) is a nonempty family of cuts bounded above by a cut \(\beta\), then \(\bigcup_{\alpha\in\mathcal A}\alpha\) is a cut and is \(\sup\mathcal A\).
Proof. The union is nonempty, downward closed, and proper because it is contained in \(\beta\). If it had a greatest rational \(q\), then \(q\in\alpha\) for some member \(\alpha\), contradicting that \(\alpha\) has no greatest element. Inclusion makes it an upper bound, and any other upper bound must contain every \(\alpha\), hence must contain the union.
Why cuts repair the rational gaps
The construction forces every bounded rational approximation process to have a target.
Remark. Dedekind completeness is order completeness. It is especially natural for monotone convergence, distribution functions, quantiles, and any argument where the endpoint is identified by what lies below it.