Dedekind cuts

The purpose of Dedekind cutsMathworldPlanetmath is to provide a sound logical foundation for the real number system. Dedekind’s motivation behind this project is to notice that a real number α, intuitively, is completely determined by the rationals strictly smaller than α and those strictly larger than α. Concerning the completeness or continuity of the real line, Dedekind notes in [2] that

If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.

Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above. He further notes that the completeness property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as cuts of rationals. Because all rationals greater than α are really just excess baggage, we prefer to sway somewhat from Dedekind’s original definition. Instead, we adopt the following definition.


A Dedekind cut is a subset α of the rational numbers that satisfies these properties:

  1. 1.

    α is not empty.

  2. 2.

    α is not empty.

  3. 3.

    α contains no greatest element

  4. 4.

    For x,y, if xα and y<x, then yα as well.

Dedekind cuts are particularly appealing for two reasons. First, they make it very easy to prove the completeness, or continuity of the real line. Also, they make it quite plain to distinguish the rationals from the irrationals on the real line, and put the latter on a firm logical foundation. In the construction of the real numbers from Dedekind cuts, we make the following definition:


A real number is a Dedekind cut. We denote the set of all real numbers by and we order them by set-theoretic inclusion, that is to say, for any α,β,

α<β if and only if αβ

where the inclusion is strict. We further define α=β as real numbers if α and β are equal as sets. As usual, we write αβ if α<β or α=β. Moreover, a real number α is said to be irrational if α contains no least element.

The Dedekind completeness property of real numbers, expressed as the supremum property, now becomes straightforward to prove. In what follows, we will reserve Greek variables for real numbers, and Roman variables for rationals.

Theorem 1.

Every nonempty subset of real numbers that is bounded above has a least upper boundMathworldPlanetmath.


Let A be a nonempty set of real numbers, such that for every αA we have that αγ for some real number γ. Now define the set


We must show that this set is a real number. This amounts to checking the four conditions of a Dedekind cut.

  1. 1.

    supA is clearly not empty, for it is the nonempty union of nonempty sets.

  2. 2.

    Because γ is a real number, there is some rational x that is not in γ. Since every αA is a subset of γ, x is not in any α, so xsupA either. Thus, supA is nonempty.

  3. 3.

    If supA had a greatest element g, then gα for some αA. Then g would be a greatest element of α, but α is a real number, so by contrapositive, supA has no greatest element.

  4. 4.

    Lastly, if xsupA, then xα for some α, so given any y<x because α is a real number yα, whence ysupA.

Thus, supA is a real number. Trivially, supA is an upper bound of A, for every αsupA. It now suffices to prove that supAγ, because γ was an arbitrary upper bound. But this is easy, because every xsupA is an element of α for some αA, so because αγ, xγ. Thus, supA is the least upper bound of A. We call this real number the supremum of A. ∎

To finish the construction of the real numbers, we must endow them with algebraic operations, define the additivePlanetmathPlanetmath and multiplicative identityPlanetmathPlanetmath elements, prove that these definitions give a field, and prove further results about the order of the reals (such as the totality of this order) – in short, build a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ordered field. This task is somewhat laborious, but we include here the appropriate definitions. Verifying their correctness can be an instructive, albeit tiresome, exercise. We use the same symbols for the operationsMathworldPlanetmath on the reals as for the rational numbers; this should cause no confusion in context.


Given two real numbers α and β, we define

  • The additive identity, denoted 0, is

  • The multiplicative identity, denoted 1, is

  • AdditionPlanetmathPlanetmath of α and β denoted α+β is

  • The opposite of α, denoted -α, is

    -α:={x:-xα, but -x is not the least element of α}
  • The absolute valueMathworldPlanetmathPlanetmathPlanetmath of α, denoted |α|, is

    |α|:={α,if α0-α,if α0
  • If α,β>0, then multiplication of α and β, denoted αβ, is

    αβ:={z:z0 or z=xy for some xα,yβ with x,y>0}

    In general,

    αβ:={0,if α=0 or β=0|α||β|if α>0,β>0 or α<0,β<0-(|α||β|)if α>0,β<0 or α>0,β<0
  • The inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of α>0, denoted α-1, is

    α-1:={x:x0 or x>0 and (1/x)α, but 1/x is not the least element of α}

    If α<0,


All that remains (!) is to check that the above definitions do indeed define a complete ordered field, and that all the sets implied to be real numbers are indeed so. The properties of as an ordered field follow from these definitions and the properties of as an ordered field. It is important to point out that in two steps, in showing that inverses and opposites are properly defined, we require an extra property of , not merely in its capacity as an ordered field. This requirement is the Archimedean property.

Moreover, because is a field of characteristicPlanetmathPlanetmathPlanetmath 0, it contains an isomorphicPlanetmathPlanetmathPlanetmath copy of . The rationals correspond to the Dedekind cuts α for which α contains a least member.


  • 1 Courant, Richard and Robbins, Herbert. What is Mathematics? pp. 68-72 Oxford University Press, Oxford, 1969
  • 2 Dedekind, Richard. Essays on the Theory of Numbers Dover Publications Inc, New York 1963
  • 3 Rudin, Walter Principles of Mathematical Analysis pp. 17-21 McGraw-Hill Inc, New York, 1976
  • 4 Spivak, Michael. Calculus pp. 569-596 Publish or Perish, Inc. Houston, 1994
Title Dedekind cuts
Canonical name DedekindCuts
Date of creation 2013-03-22 12:38:34
Last modified on 2013-03-22 12:38:34
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 26
Author rmilson (146)
Entry type Definition
Classification msc 26A03
Synonym Schnitt
Related topic RealNumber