Dedekind Cuts

Dedekind cuts provide one of the most elegant constructions of the real numbers from the rationals. The idea, due to Richard Dedekind (1872), is to define a real number as a "partition" of $\mathbb{Q}$ into a lower set and an upper set. This approach makes completeness manifest: every cut automatically corresponds to a unique real number.

Definition and examples

Definition1.1Dedekind cut

A Dedekind cut is a subset $\alpha \subseteq \mathbb{Q}$ satisfying:

$\alpha \neq \varnothing$ and $\alpha \neq \mathbb{Q}$ .
If $p \in \alpha$ and $q < p$ , then $q \in \alpha$ (downward closed).
$\alpha$ has no greatest element: if $p \in \alpha$ , there exists $r \in \alpha$ with $r > p$ .

We denote the set of all Dedekind cuts by $\mathbb{R}$ .

RemarkIntuition

A cut $\alpha$ represents the "lower half" of $\mathbb{Q}$ corresponding to some real number. The cut has no largest element (condition 3) to avoid ambiguity: the number $\sqrt{2}$ is defined by the set $\{q \in \mathbb{Q} \mid q < \sqrt{2}\}$ , not by adjoining $\sqrt{2}$ itself. This ensures each real number corresponds to exactly one cut.

ExampleCuts corresponding to rationals

For $r \in \mathbb{Q}$ , define

$r^* = \{q \in \mathbb{Q} \mid q < r\}.$

Then $r^*$ is a Dedekind cut. The map $r \mapsto r^*$ embeds $\mathbb{Q}$ into $\mathbb{R}$ (we identify $r$ with $r^*$ ). Note that $r^*$ has no largest element, even though $r$ is rational — the cut is the set of rationals strictly less than $r$ .

ExampleThe cut for √2

Define

$\alpha = \{p \in \mathbb{Q} \mid p \leq 0 \text{ or } p^2 < 2\}.$

Then $\alpha$ is a Dedekind cut. It represents the irrational number $\sqrt{2}$ . To verify condition (3): if $p \in \alpha$ with $p > 0$ , then $p^2 < 2$ , so we can find $r$ slightly larger than $p$ with $r^2 < 2$ still (by choosing $r = p + \epsilon$ for small enough $\epsilon > 0$ ).

ExampleThe cut for π

Similarly,

$\pi^* = \{q \in \mathbb{Q} \mid q < \pi\}$

is the Dedekind cut corresponding to $\pi$ . Since $\pi$ is irrational, this cut has no rational "boundary" — the rationals in $\pi^*$ get arbitrarily close to $\pi$ but never reach it.

ExampleCuts have no largest element

The set $\beta = \{q \in \mathbb{Q} \mid q \leq 2\}$ is not a Dedekind cut: it violates condition (3) because $2$ is the greatest element of $\beta$ . To represent the real number $2$ , we use the cut $2^* = \{q \in \mathbb{Q} \mid q < 2\}$ , which has no largest element.

Order on Dedekind cuts

Definition1.2Order on R

For Dedekind cuts $\alpha, \beta \in \mathbb{R}$ , define

$\alpha < \beta \iff \alpha \subsetneq \beta \quad \text{($\alpha$ is a proper subset of $\beta$)}.$

RemarkTotal order

This relation $<$ makes $\mathbb{R}$ a totally ordered set: for any $\alpha, \beta \in \mathbb{R}$ , exactly one of $\alpha < \beta$ , $\alpha = \beta$ , or $\beta < \alpha$ holds. The ordering is consistent with the usual ordering on $\mathbb{Q}$ : $r < s$ in $\mathbb{Q}$ iff $r^* < s^*$ in $\mathbb{R}$ .

ExampleOrdering involving √2

Let $\alpha$ be the cut for $\sqrt{2}$ (from the previous example), and let $r = 3/2 \in \mathbb{Q}$ . Then $\alpha < r^*$ because $\alpha \subsetneq r^*$ : every rational less than $\sqrt{2}$ is also less than $3/2$ , but $3/2 \notin \alpha$ (since $(3/2)^2 = 9/4 > 2$ ), so $\alpha$ is strictly smaller.

Arithmetic operations

Definition1.3Addition of cuts

For Dedekind cuts $\alpha, \beta \in \mathbb{R}$ , define

$\alpha + \beta = \{r + s \mid r \in \alpha, \, s \in \beta\}.$

RemarkWell-defined

It can be verified that $\alpha + \beta$ is again a Dedekind cut. The additive identity is $0^* = \{q \in \mathbb{Q} \mid q < 0\}$ . The additive inverse of $\alpha$ is defined carefully to ensure $\alpha + (-\alpha) = 0^*$ .

Definition1.4Multiplication of positive cuts

For positive cuts $\alpha, \beta > 0^*$ , define

$\alpha \cdot \beta = \{p \in \mathbb{Q} \mid p \leq rs \text{ for some } r \in \alpha, s \in \beta \text{ with } r, s > 0\}.$

Multiplication is extended to all cuts by cases (using signs).

RemarkField structure

With these operations, $\mathbb{R}$ becomes a field, extending $\mathbb{Q}$ . The verification that $\mathbb{R}$ is an ordered field (with the subset order defined above) is a standard exercise in real analysis.

ExampleAdding cuts

Let $\alpha$ be the cut for $\sqrt{2}$ and $\beta = 1^*$ (the cut for $1$ ). Then $\alpha + \beta$ is the cut for $\sqrt{2} + 1$ :

$\alpha + \beta = \{r + s \mid r^2 < 2 \text{ (or } r \leq 0), \, s < 1\}.$

This is the set of all rationals less than $\sqrt{2} + 1$ .

Completeness of Dedekind cuts

The key theorem is that $\mathbb{R}$ , constructed via Dedekind cuts, is complete.

Theorem1.1Completeness of R

Every nonempty set $A \subseteq \mathbb{R}$ that is bounded above has a least upper bound (supremum) in $\mathbb{R}$ .

Proof

Let $A \subseteq \mathbb{R}$ be nonempty and bounded above. Define

$\gamma = \bigcup_{\alpha \in A} \alpha.$

In other words, $\gamma$ consists of all rationals $q$ such that $q \in \alpha$ for some $\alpha \in A$ .

Claim: $\gamma$ is a Dedekind cut, and $\gamma = \sup A$ .

$\gamma \neq \varnothing$ (since $A$ is nonempty).
$\gamma \neq \mathbb{Q}$ (since $A$ is bounded above by some cut $\beta$ , so no $\alpha \in A$ contains all rationals in $\beta$ 's complement).
Downward closed: if $q \in \gamma$ , then $q \in \alpha$ for some $\alpha \in A$ . If $p < q$ , then $p \in \alpha$ (since $\alpha$ is downward closed), hence $p \in \gamma$ .
No greatest element: if $q \in \gamma$ , then $q \in \alpha$ for some $\alpha \in A$ . Since $\alpha$ has no greatest element, there exists $r \in \alpha$ with $r > q$ . Thus $r \in \gamma$ and $r > q$ .

Therefore, $\gamma$ is a cut. It is clearly an upper bound of $A$ (since $\alpha \subseteq \gamma$ for all $\alpha \in A$ ), and it is the least upper bound: any upper bound $\beta$ must satisfy $\alpha \subseteq \beta$ for all $\alpha \in A$ , hence $\gamma = \bigcup \alpha \subseteq \beta$ .

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RemarkUniqueness of R

The completeness property, combined with being an ordered field containing $\mathbb{Q}$ as a dense subset, uniquely characterizes $\mathbb{R}$ up to isomorphism. Any two complete ordered fields are isomorphic. This is why Dedekind's construction and the Cauchy sequence construction yield the "same" real numbers.

Comparison with Cauchy sequences

An alternative construction of $\mathbb{R}$ uses equivalence classes of Cauchy sequences of rationals. Both constructions produce the same complete ordered field.

ExampleCauchy sequences vs. cuts

The decimal expansion $1.414213\ldots$ of $\sqrt{2}$ corresponds to:

Cauchy sequence: $(1.4, 1.41, 1.414, 1.4142, \ldots)$ in $\mathbb{Q}$ .
Dedekind cut: $\alpha = \{q \in \mathbb{Q} \mid q^2 < 2 \text{ or } q \leq 0\}$ .

Both encode the same real number. The Cauchy approach emphasizes approximation; the Dedekind approach emphasizes partitioning the rationals.

RemarkPhilosophical note

Dedekind's construction is set-theoretic and static: a real number "is" a subset of $\mathbb{Q}$ . The Cauchy construction is analytic and dynamic: a real number "is" a limit of a sequence. Both are valid; the choice is a matter of taste and context.

Summary

Dedekind cuts provide a rigorous construction of $\mathbb{R}$ from $\mathbb{Q}$ :

A Dedekind cut is a downward-closed subset of $\mathbb{Q}$ with no greatest element.
The set $\mathbb{R}$ of all cuts forms a complete ordered field.
Completeness is immediate: the union of cuts is again a cut, giving the supremum.

This construction makes the least upper bound property manifest and serves as a cornerstone for developing real analysis. See Completeness Axiom for further consequences.