Dedekind cut

A Dedekind cut is in the mathematical order theory is a special partition of the rational numbers with the help of a real number can be represented. In this way, one can construct the real numbers from the rational numbers. It is named this "method of Dedekind cuts " after the German mathematician Richard Dedekind. It can be widely used for completing orders, which are like the rational numbers dense in itself. Also in this method, the generalization of the common designations are defined and used in this article.

If we define the axiomatic real numbers, so you can use Dedekind cuts to secure the order completeness of the real numbers. In this case one speaks of the Axiom of Dedekind cut or cut short by the axiom.

Definition

Dedekind cuts are an ordered pair of subsets of rational numbers ( subset ) and ( superset of ) the following axioms defined:

Since each identifying the subset or superset for themselves a cut, you can also use the following definition:

A subset of the rational numbers if and only subset of a Dedekind section if the following conditions are met:

These three conditions can be summarized formulated as follows: is an open, unrestricted by downward and upward bounded interval of rational numbers. Instead of " subset of a Dedekind cut " in the literature the term "open top " used. Sometimes the subset of a Dedekind cut is also described himself as " average ".

Construction of the real numbers

One defines the set of real numbers as the set of all ( Dedekind ) cuts in. For simplicity, in the following, as only the subsets of Dedekind cuts described above considered and referred to as "cuts". In the set of all sections to embed the rational numbers by assigning each number as cut the set of all smaller numbers. So the rational number one assigns the average

About. But the irrational numbers can be represented by cuts. The number corresponds to, for example, the average

This may be called the cuts make sense "numbers", you have to set the arithmetic operations and the order of the new numbers so that they continue the arithmetic operations on the rational numbers and their order.

Be thereto and any two sections.

Order

It is precisely when proper subset of is.

This defines a strict total order. This is even order complete ( by construction ), that is every bounded subset has a supremum. So Thus, if a lot of cuts and an upper bound, then every cut is a subset of. The union of all is then a cut, the least upper bound of.

Addition

One defines.

One can show that this is indeed an addition, ie a commutative, associative link, defined and that there is for each section, an additive inverse element. Furthermore, the definition of this addition coincides with the addition of already known together.

Multiplication

For and we define the multiplication as follows:

This multiplication can be extended to all by

And

Defined. Also, this multiplication is associative, commutative and there is each an inverse. In addition, this multiplication falls with the on together, if the factors are rational.

Generalizations

• Applying the construction of Dedekind cuts back on the ordered set, so there are no new elements, each section is created by an associated section number. This property is also referred to as the intersection axiom and is almost literally equivalent to Supremumsaxiom.

• Each ( in itself ) density strict total ordering (M, <) can be embedded (on M instead ) into a proper full order N with the help of Dedekind cuts. In terms of the theory of order a totally ordered set is dense ordered in when between two different elements is always a third. Whether and how other existing structures on M ( as below on the links of addition and multiplication ) can be " useful " to continue on N depends on the specific application. → Compare this order topology.

• One of the Dedekind cuts very similar method is used for constructing the surreal numbers.