Dedekind cut

thumb| right| 350px| Dedekind used his cut to construct the [[irrational number|irrational, real numbers.]]

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of constructing the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two nonempty sets A and B, such that each element of A is less than every element of B, and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.

This cut represents the irrational number <math>\sqrt{2}</math> in Dedekind's construction. The essential idea is that we use a set <math>A</math>, which is the set of all rational numbers whose squares are less than 2, to "represent" the number <math>\sqrt{2}</math>, 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 <math>A</math> really is a cut (according to the definition) and the square of <math>A</math>, that is <math>A \times A</math> (please refer to the link above for the precise definition of how the multiplication of cuts is defined), is <math>2</math> (note that rigorously speaking this number 2 is represented by a cut <math>\{x\ |\ x \in \mathbb{Q}, x < 2\}</math>). To show the first part, we show that for any positive rational <math>x</math> with <math>x^2 < 2</math>, there is a rational <math>y</math> with <math>x < y</math> and <math>y^2 < 2</math>. The choice <math>y=\frac{2x+2}{x+2}</math> works, thus <math>A</math> is indeed a cut. Now armed with the multiplication between cuts, it is easy to check that <math>A \times A \le 2</math> (essentially, this is because <math>x \times y \le 2, \forall x, y \in A, x, y \ge 0</math>). Therefore, to show that <math>A \times A = 2</math>, we show that <math>A \times A \ge 2</math>, and it suffices to show that for any <math>r < 2</math>, there exists <math>x \in A</math>, <math>x^2 > r</math>. For this we notice that if <math>x > 0, 2-x^2=\epsilon > 0</math>, then <math>2-y^2 \le \frac{\epsilon}{2}</math> for the <math>y</math> constructed above, this means that we have a sequence in <math>A</math> whose square can become arbitrarily close to <math>2</math>, which finishes the proof.

Note that the equality cannot hold since <math>\sqrt{2}</math> is not rational.

Relation to interval arithmetic

Given a Dedekind cut representing the real number <math>r</math> by splitting the rationals into <math>(A,B)</math>

where rationals in <math>A</math> are less than <math>r</math> and rationals in <math>B</math> are greater than <math>r</math>, it can be equivalently represented as the set of pairs <math>(a,b)</math> with <math>a \in A</math> and <math>b \in B</math>, with the lower cut and the upper cut being given by projections. This corresponds exactly to the set of intervals approximating <math>r</math>.

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 <math>A</math> and <math>B</math> 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 <math>(A,B)</math> such that <math>A \cup B = X </math> and <math>a \in A</math>, <math>b \in B</math> imply <math>a < b</math>. 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 .

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.

Notes

References

• Dedekind, Richard, Essays on the Theory of Numbers, "Continuity and Irrational Numbers," Dover Publications: New York, . Also available at Project Gutenberg.

External links