In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements 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. Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards and B is closed upwards, and A contains no greatest element. See also completeness. It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number. In other words, the number line where every real number is defined as a Dedekind cut of rationals is a completecontinuum without any further gaps.
Definition
A Dedekind cut is a partition of the rationals into two subsets A and B such that
It is more symmetrical to use the 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 of S, the set B must have a minimal elementb, hence we must have that A is the interval. In this case, we say that bis represented by the cut. The important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the original collection of 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 by putting every negative rational number in A, along with every non-negative 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, 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 as less than another Dedekind cut if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut is again less than. In this way, set inclusion can be used to represent the ordering of numbers, and all other relations can be similarly created from set relations. The set of all Dedekind cuts is itself a linearly ordered set. 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 linearly ordered set that does have this useful property.
Construction of the real numbers
A typical Dedekind cut of the rational numbers is given by the partition with This cut represents the irrational number in Dedekind's construction. The essential idea is that we use a set, which is the set of all rational numbers whose squares are less than 2, to "represent" number, and further, by defining properly arithmetic operators over these sets, these sets form the familiar real numbers. To establish this, one must show that really is a cut and the square of, that is , is . To show the first part, we show that for any positive rational with, there is a rational with and. The choice works, thus is indeed a cut. Now armed with the multiplication between cuts, it is easy to check that . Therefore to show that, we show that, and it suffices to show that for any, there exists,. For this we notice that if, then for the constructed above, this means that we have a sequence in whose square can become arbitrarily close to, which finishes the proof. Note that the equality cannot hold since is not rational.
Generalizations
A construction similar to Dedekind cuts is used for the construction of surreal numbers.
More generally, if S is a partially ordered set, a completion of S means a complete latticeL 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 Au denote the set of upper bounds of A, and let Al denote the set of lower bounds of A. Then the Dedekind–MacNeille completion of S consists of all subsets A for which l = A; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it.
