Dedekind cut

In mathematical order theory, a Dedekind cut is a special partition of rational numbers that can be used to represent a real number . In this way one can construct the real numbers from the rational numbers. This “method of Dedekind cuts” is named after the German mathematician Richard Dedekind , although such partitions were first described by Joseph Bertrand , as Detlef Spalt discovered. It can generally be used to complete orders which, like rational numbers, are inherently close. Even with this generalization of the method, the terms that are defined and used in this article are common.

If one defines the real numbers axiomatically, then one can use Dedekind cuts to ensure the order completeness of the real numbers. In this case one speaks of the axiom of the Dedekind cut or, for short, of the cut axiom .

definition

Dedekind cuts are defined by an ordered pair of subsets of rational numbers (subset) and (superset) using the following axioms: ${\displaystyle \alpha }$ ${\displaystyle \beta }$

1. Every rational number is in exactly one of the sets , . ${\displaystyle \alpha }$ ${\displaystyle \beta }$
2. Neither nor is it empty. ${\displaystyle \alpha }$ ${\displaystyle \beta }$
3. Each element of is smaller than each element of . ${\displaystyle \alpha }$ ${\displaystyle \beta }$
4. ${\displaystyle \alpha }$has no largest element , that is, for each there is one with . ${\displaystyle p\in \alpha }$ ${\displaystyle r\in \alpha }$ ${\displaystyle p<r}$

Since the subset or the superset each define a section, the following definition can also be used: ${\displaystyle \alpha }$ ${\displaystyle \beta }$

A subset of the rational numbers is a subset of a Dedekind cut if and only if the following conditions are met: ${\displaystyle \alpha }$

1. ${\displaystyle \alpha }$is not empty and does not include all rational numbers ( ). ${\displaystyle \alpha \neq \mathbb {Q} }$
2. ${\displaystyle \alpha }$is closed at the bottom, that is, if , and , then is also . ${\displaystyle p\in \alpha }$ ${\displaystyle q\in \mathbb {Q} }$ ${\displaystyle p>q}$ ${\displaystyle q\in \alpha }$
3. ${\displaystyle \alpha }$ does not contain the largest element.

These three conditions can be summarized as follows: is an open, downwardly unbounded and upwardly bounded interval of rational numbers. Instead of “subset of a Dedekindian cut”, the term “open beginning” is also used in the literature. Sometimes the subset of a Dedekind cut is referred to as a "cut" itself. ${\displaystyle \alpha }$

Construction of the real numbers

The set of real numbers is defined as the set of all (Dedekindian) cuts in . For the sake of simplicity, in the following, as described above, only the subsets of Dedekind cuts are considered and referred to as “cuts”. The rational numbers are embedded in the set of all cuts by assigning the set of all smaller numbers to each number as the cut. So one assigns the cut to the rational number ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle x\in \mathbb {Q} }$

${\displaystyle x^{*}:=\{s\in \mathbb {Q} \mid s<x\}}$

to. But the irrational numbers can also be represented by cuts. For example, the number corresponds to the cut ${\displaystyle {\sqrt {2}}}$

${\displaystyle \{s\in \mathbb {Q} \mid s<0{\text{ oder }}s^{2}<2\}\,}$.

In order to be able to call the cuts meaningfully “numbers”, one must fix the arithmetic operations and the order of the new numbers in such a way that they continue the arithmetic operations on the rational numbers and their order.

Add to this and any two cuts. ${\displaystyle \alpha }$ ${\displaystyle \beta }$

order

One sets if and only if is a real subset of . ${\displaystyle \alpha <\beta }$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$

This defines a strict total ordering on . This is even (according to construction) orderly complete , i.e. every limited subset has a supremum . Namely, if there is a set of cuts and an upper bound , then every cut is a subset of . The union of all is then also a cut, the smallest upper bound of . ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle \beta }$ ${\displaystyle \alpha \in A}$ ${\displaystyle \beta }$ ${\displaystyle \alpha \in A}$ ${\displaystyle A}$

addition

One defines . ${\displaystyle \alpha +\beta :=\{r+s\mid r\in \alpha ,s\in \beta \}}$

One can show that this actually defines an addition, i.e. a commutative, associative connection, and that there is an additively inverse element for every intersection . Furthermore, the definition of this addition coincides with the already known addition . ${\displaystyle \alpha }$ ${\displaystyle -\alpha }$ ${\displaystyle \mathbb {Q} }$

multiplication

For and one defines the multiplication as follows: ${\displaystyle \alpha >0^{*}}$ ${\displaystyle \beta >0^{*}}$

${\displaystyle \alpha \cdot \beta :=\{p\in \mathbb {Q} \mid \exists \,r\in \alpha ,s\in \beta ,r,s>0\colon p\leq r\cdot s\}}$

This multiplication can be extended to the full by adding ${\displaystyle \mathbb {R} }$

${\displaystyle \alpha \cdot 0^{*}:=0^{*}\cdot \alpha :=0^{*}}$

and

${\displaystyle \alpha \cdot \beta :={\begin{cases}(-\alpha )\cdot (-\beta )&\alpha ,\beta <0^{*}\\-((-\alpha )\cdot (\beta ))&\alpha <0^{*},\beta >0^{*}\\-((\alpha )\cdot (-\beta ))&\alpha >0^{*},\beta <0^{*}\end{cases}}}$

Are defined. This multiplication is also associative, commutative and there is an inverse to each . In addition, this multiplication also coincides with the one if the factors are rational. ${\displaystyle a\neq 0}$ ${\displaystyle a^{-1}}$ ${\displaystyle \mathbb {Q} }$

Generalizations

• If the construction of Dedekind cuts is applied again to the ordered set , no new elements are created, each cut is created by an associated number of cuts. This property is also known as the axiom of intersection and is almost literally equivalent to the axiom of the supreme . ${\displaystyle (\mathbb {R} ,<)}$

• Every (in itself) dense strict total order (M, <) can be embedded in a complete order N with the help of Dedekindian cuts (on M instead of ) . In the sense of order theory, a totally ordered set is tightly ordered if there is always a third between two different elements. Whether and how other structures existing on M (such as the links addition and multiplication here) can be continued “sensibly” on N depends on the specific application (compare order topology ). ${\displaystyle \mathbb {Q} }$

• A method very similar to Dedekind's cuts is used to construct the surreal numbers .

See also

• Cauchy episode

literature

• Joseph Bertrand: Traité d'Arithmétique. Crapelet, Paris 1849. ( online ).
• Richard Dedekind: Continuity and Irrational Numbers. Friedrich Vieweg and son, Braunschweig 1872. ( online ).
• Oliver Deiser: Basic Concepts in Scientific Mathematics. Springer 2010, ISBN 978-3-642-11488-5 , pp. 118-120 ( excerpt (Google) ).
• K. Mainzer Real Numbers. Chapter 2 (Paragraph 2 on Dedekind cuts) in: Heinz-Dieter Ebbinghaus u. a .: Numbers. , Springer Verlag 1983, p. 30 f.
• Harro Heuser : Textbook of Analysis. Part 1. Vieweg + Teubner, Wiesbaden 1980, 6th updated edition. ibid. 1988, ISBN 3-519-42221-2 , pp. 29-32, 36-38

Web links

• Dedekind cut in the Encyclopedia of Mathematics
• Dedekind cut on PlanetMath
• A. Bogomolny: On Dedekind Cuts on cut-the-knot.org
• Dedekind cut in the MacTutor History of Mathematics archive
• Construction of ${\displaystyle \mathbb {R} }$ (PDF; 65 kB) - part of an analysis script from the University of Münster

References and comments

1. ↑ Joseph Bertrand: Traité d'Arithmétique 1849, p. 203: “An incommensurable number can only be defined by indicating how the quantity it expresses can be formed by the unit. In the following we assume that this definition consists in indicating which commensurable numbers are smaller or larger than the number .... "
2. ↑ Detlef Spalt: A Brief History of Analysis . Springer, 2019, p. 229, doi : 10.1007 / 978-3-662-57816-2 .
3. ↑ This means that represent a decomposition (partitioning) of the set of rational numbers . It is . ${\displaystyle (\alpha ,\beta )}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \beta =\mathbb {Q} \setminus \alpha }$
4. ^ Fritz Reinhardt, Heinrich Soeder: dv-Atlas for mathematics. Deutscher Taschenbuchverlag, Munich 1974, page 59.
5. ^ Edmund Landau: Fundamentals of Analysis. Akademische Verlagsgesellschaft MBH, Leipzig 1930, Chapter 3, Section 1, Definition 28.
6. ^ Walter Rudin: Analysis. Oldenbourg Verlag, Munich 2005, ISBN 978-3-486-57852-2 , page 19.
7. ↑ The example can easily be generalized: The number with rational corresponds to the cut ${\displaystyle {\sqrt {x}}}$ ${\displaystyle x>0}$

   ${\displaystyle \{s\in \mathbb {Q} \mid s<0{\text{ oder }}s^{2}<x\}}$.

   If is rational, this falls back on the definition (for ) above . ${\displaystyle {\sqrt {x}}}$ ${\displaystyle {\sqrt {x}}^{*}}$