Dedekind cut

Representations

It is more symmetrical to use the (A, B) notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward-closed set A without greatest element a "Dedekind cut".

If the ordered set S is complete, then, for every Dedekind cut (A, B) of S, the set B must have a minimal element b, so that A is the interval (−∞, b), and B the interval [b, +∞). In this case, we say that b is represented by the cut (A, B).

The main purpose of the Dedekind cut is to allow us to work with number sets that are not complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers). The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents.

For example if A and B only contain rational numbers, they can still be cut at ${\displaystyle \sqrt{2}}$ by putting every negative rational number in A, along with every non-negative rational number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for ${\displaystyle \sqrt{2}}$, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number.

Ordering of cuts

Regard one Dedekind cut (A, B) as less than another Dedekind cut (C, D) (of the same superset) if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut (A, B) is again less than (C, D). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (greater than, less than or equal to, equal to, and so on) can be similarly created from set relations.

The set of all Dedekind cuts is itself a linearly ordered set (of sets). Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set S, which might not have had the least-upper-bound property, within a (usually larger) linearly ordered set that does have this useful property.

Construction of the real numbers

A typical Dedekind cut of the rational numbers ${\displaystyle \mathbb{Q}}$ is given by the partition ${\displaystyle (A,B)}$ with

${\displaystyle A=\{a\in \mathbb{Q}:a^2<2\text{ or }a<0\},}$

${\displaystyle B=\{b\in \mathbb{Q}:b^2\ge 2\text{ and }b\ge 0\}.}$

This cut represents the irrational number ${\displaystyle \sqrt{2}}$ in Dedekind's construction. The essential idea is that we use a set ${\displaystyle A}$, which is the set of all rational numbers whose squares are less than 2, to "represent" the number ${\displaystyle \sqrt{2}}$, and further, by defining properly arithmetic operators over these sets (addition, subtraction, multiplication, and division), these sets (together with these arithmetic operations) form the familiar field of real numbers.

To establish this, one must show that ${\displaystyle A}$ really is a cut (according to the definition) and the square of ${\displaystyle A}$, that is ${\displaystyle A\times A}$ (please refer to the link above for the precise definition of how the multiplication of cuts is defined), is ${\displaystyle 2}$ (note that rigorously speaking this number 2 is represented by a cut ${\displaystyle \{x|x\in \mathbb{Q},x<2\}}$). To show the first part, we show that for any positive rational ${\displaystyle x}$ with ${\displaystyle x^2<2}$, there is a rational ${\displaystyle y}$ with ${\displaystyle x<y}$ and ${\displaystyle y^2<2}$. The choice ${\displaystyle y=\frac{2x+2}{x+2}}$ works, thus ${\displaystyle A}$ is indeed a cut. Now armed with the multiplication between cuts, it is easy to check that ${\displaystyle A\times A\le 2}$ (essentially, this is because ${\displaystyle x\times y\le 2,\mathrm{\forall }x,y\in A,x,y\ge 0}$). Therefore, to show that ${\displaystyle A\times A=2}$, we show that ${\displaystyle A\times A\ge 2}$, and it suffices to show that for any ${\displaystyle r<2}$, there exists ${\displaystyle x\in A}$, ${\displaystyle x^2>r}$. For this we notice that if ${\displaystyle x>0,2-x^2=ϵ>0}$, then ${\displaystyle 2-y^2\le \frac{ϵ}{2}}$ for the ${\displaystyle y}$ constructed above, this means that we have a sequence in ${\displaystyle A}$ whose square can become arbitrarily close to ${\displaystyle 2}$, which finishes the proof.

Note that the equality b² = 2 cannot hold since ${\displaystyle \sqrt{2}}$ is not rational.

Relation to interval arithmetic

Given a Dedekind cut representing the real number ${\displaystyle r}$ by splitting the rationals into ${\displaystyle (A,B)}$ where rationals in ${\displaystyle A}$ are less than ${\displaystyle r}$ and rationals in ${\displaystyle B}$ are greater than ${\displaystyle r}$, it can be equivalently represented as the set of pairs ${\displaystyle (a,b)}$ with ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$, with the lower cut and the upper cut being given by projections. This corresponds exactly to the set of intervals approximating ${\displaystyle r}$.

This allows the basic arithmetic operations on the real numbers to be defined in terms of interval arithmetic. This property and its relation with real numbers given only in terms of ${\displaystyle A}$ and ${\displaystyle B}$ is particularly important in weaker foundations such as constructive analysis.

Generalizations

Arbitrary linearly ordered sets

In the general case of an arbitrary linearly ordered set X, a cut is a pair ${\displaystyle (A,B)}$ such that ${\displaystyle A\cup B=X}$ and ${\displaystyle a\in A}$, ${\displaystyle b\in B}$ imply ${\displaystyle a<b}$. Some authors add the requirement that both A and B are nonempty.

If neither A has a maximum, nor B has a minimum, the cut is called a gap. A linearly ordered set endowed with the order topology is compact if and only if it has no gap.

Surreal numbers

A construction resembling Dedekind cuts is used for (one among many possible) constructions of surreal numbers. The relevant notion in this case is a Cuesta-Dutari cut, named after the Spanish mathematician Norberto Cuesta Dutari.

Partially ordered sets

More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals.

One completion of S is the set of its downwardly closed subsets, ordered by inclusion. A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of S, let A^u denote the set of upper bounds of A, and let A^l denote the set of lower bounds of A. (These operators form a Galois connection.) Then the Dedekind–MacNeille completion of S consists of all subsets A for which (A^u)^l = A; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it.