Editing Dedekind cut

{{short description|Method of construction of the real numbers}}
{{For|the American record producer known professionally as Dedekind Cut|Fred Warmsley}}
[[File:Dedekind cut at square root of two.svg| thumb| right| 350px| Dedekind used his cut to construct the [[irrational number|irrational]], [[real number]]s.]]

In [[mathematics]], '''Dedekind cuts''', named after German mathematician [[Richard Dedekind]] (but previously considered by [[Joseph Bertrand]]<ref>{{cite book|last=Bertrand|first=Joseph|title=Traité d'Arithmétique |url = https://gallica.bnf.fr/ark:/12148/bpt6k77735p/f209.image.r=%22joseph%20bertrand%22 |year=1849|at=page 203|quote=An incommensurable number can be defined only by indicating how the magnitude it expresses can be formed by means of unity. In what follows, we suppose that this definition consists of indicating which are the commensurable numbers smaller or larger than it ....}}</ref><ref>{{cite book |last=Spalt |first=Detlef |title=Eine kurze Geschichte der Analysis|year=2019|publisher=Springer|doi=10.1007/978-3-662-57816-2|isbn=978-3-662-57815-5 }}</ref>), are а method of [[construction of the real numbers]] from the [[rational number]]s. A Dedekind cut is a [[partition of a set|partition]] of the rational numbers into two [[Set (mathematics) |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&nbsp;''B''.<ref name=":0">{{cite book|last=Dedekind |first=Richard |title=Continuity and Irrational Numbers |url=http://www.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf#page=15 |year=1872|at=Section IV |quote=Whenever, then, we have to do with a cut produced by no rational number, we create a new ''irrational'' number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....}}</ref> 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.<ref name=":0" />

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 (meaning that for all ''a'' in ''A'', ''x'' ≤ ''a'' implies that ''x'' is in ''A'' as well) and ''B'' is closed upwards, and ''A'' contains no greatest element. See also [[completeness (order theory)]].

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 (which can be identified as the smallest element of the ''B'' set). In other words, the [[number line]] where every [[real number]] is defined as a Dedekind cut of rationals is a [[Complete metric space|complete]] [[linear continuum|continuum]] without any further gaps.

== Definition ==

A Dedekind cut is a partition of the rationals <math>\mathbb{Q}</math> into two subsets <math>A</math> and <math>B</math> such that

# <math>A</math> is nonempty.
# <math>A \neq \mathbb{Q}</math> (equivalently, <math>B</math> is nonempty).
# If <math>x, y \in \mathbb{Q}</math>, <math> x < y </math>, and <math> y \in A </math>, then <math> x \in A </math>. (<math>A</math> is "closed downwards".)
# If <math> x \in A </math>, then there exists a <math> y \in A </math> such that <math> y > x </math>. (<math>A</math> does not contain a greatest element.)

By omitting  the first two requirements, we formally obtain the [[extended real number line]].

== 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'', 
hence we must have that  ''A'' is the [[interval (mathematics)|interval]] (&minus;∞, ''b''), and ''B'' the interval [''b'', +∞).
In this case, we say that ''b'' ''is represented by'' the cut (''A'', ''B'').

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 (most often [[rational number]]s). 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 <math>\sqrt{2}</math> 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 <math>\sqrt{2}</math>, 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==
{{See also|Construction of the real numbers#Construction by Dedekind cuts}}
A typical Dedekind cut of the [[rational number]]s <math>\Q</math> is given by the partition <math>(A,B)</math> with

:<math>A = \{ a\in\mathbb{Q} : a^2 < 2 \text{ or } a < 0 \},</math>
:<math>B = \{ b\in\mathbb{Q} : b^2 \ge 2 \text{ and } b \ge 0 \}.</math><ref>In the second line, <math>\ge</math> may be replaced by <math>></math> without any difference as there is no solution for <math>x^2 = 2</math> in <math>\Q</math> and <math>b=0</math> is already forbidden by the first condition. This results in the equivalent expression
:<math>B = \{ b\in\mathbb{Q} : b^2 > 2 \text{ and } b > 0 \}.</math></ref>

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" 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 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 {{math|1=''b''<sup>2</sup>&nbsp;=&nbsp;2}} cannot hold since [[Square root of 2#Proofs of irrationality|<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.<ref>R. Engelking, General Topology, I.3</ref>

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.<ref>Jun-Iti Nagata, Modern General Topology, Second revised edition, Theorem VIII.2, p. 461. Actually, the theorem holds in the setting of generalized ordered spaces, but in this more general setting pseudo-gaps should be taken into account.</ref>

===Surreal numbers===
A construction resembling Dedekind cuts is used for (one among many possible) constructions of [[surreal number]]s. The relevant notion in this case is a Cuesta-Dutari cut,<ref name="Alling">{{cite book | last = Alling | first = Norman L. | title = Foundations of Analysis over Surreal Number Fields | publisher = North-Holland | series = Mathematics Studies 141 | year = 1987 | isbn = 0-444-70226-1}}</ref> named after the Spanish mathematician {{Ill|Norberto Cuesta Dutari|es}}.

===Partially ordered sets===
{{Main|Dedekind–MacNeille completion}}
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 [[subset|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''<sup>u</sup> denote the set of upper bounds of ''A'', and let ''A''<sup>l</sup> 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''<sup>u</sup>)<sup>l</sup> = ''A''; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with ''S'' embedded in it.

==Notes==
{{reflist}}

==References==
*Dedekind, Richard, ''Essays on the Theory of Numbers'', "Continuity and Irrational Numbers," Dover Publications: New York, {{ISBN|0-486-21010-3}}. Also [https://www.gutenberg.org/ebooks/21016 available] at Project Gutenberg.

==External links==
* {{springer|title=Dedekind cut|id=p/d030530}}


{{Rational numbers}}
 
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