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In mathematics, a 'Dedekind cut', named after Richard Dedekind, in a totally ordered set ''S'' is a partition of it, (''A'', ''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. The ''cut'' itself is, conceptually, the "gap" defined between ''A'' and ''B''. The original and most important cases are Dedekind cuts for rational numbers and real numbers. Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom). See also completeness (order theory).
The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is ''created'' by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity.
Dedekind used the ambiguous word cut (Schnitt) in the geometric sense. That is, it is an intersection of a line with another line that crosses it. It is not a gap. When one line crosses another in geometry, it is said to cut that line. In this case, one of the lines is the number line. Both lines have one point in common. At that one point on the number line, if there is no rational number, the mathematician posits or arbitrarily places an irrational number. This results in the positioning of a real number at every point on the continuum.

Contents

Handling Dedekind cuts

Ordering Dedekind cuts

The cut construction of the real numbers

Additional structure on the cuts

Generalization: Dedekind completions in posets

Dedekind-MacNeille completion

Another generalization: surreal numbers

Allusions

Bibliography

References

See also

Handling Dedekind cuts

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 last element a "Dedekind cut".
If the ordered set ''S'' is complete, then every set ''B'' in a Dedekind cut (''A'', ''B'') must have a minimal element ''b'',
hence we must have that ''A'' is the interval ( −∞, ''b''), and ''B'' the interval [''b'', +∞).

Ordering Dedekind cuts

Regard one Dedekind cut { ''A'', ''B'' } as ''less than'' another Dedekind cut { ''C'', ''D'' } if ''A'' is a proper subset of ''C'', or, equivalently ''D'' is a proper subset of ''B''. In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it has the least-upper-bound property, i.e., every nonempty subset of it that has an upper bound has a ''least'' upper bound. Embedding ''S'' within a larger linearly ordered set that does have the least-upper-bound property is the purpose.

The cut construction of the real numbers

The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by
:
:$B\; =\; \{\; binmathbb\{Q\}\; :\; b^2\; ge\; 2\; land\; b\; >\; 0\; \}.$
This cut represents the real number $sqrt\{2\}$ in Dedekind's construction. Note that the equality $b^2\; =\; 2$ cannot hold since that would imply that$sqrt\{2\}$ is rational.

Additional structure on the cuts

''See construction of real numbers''

Generalization: Dedekind completions in posets

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 (also called order ideals), ordered by inclusion. ''S'' is embedded in this lattice by sending each element ''x'' to the ideal it generates.

Dedekind-MacNeille completion

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''^u denote the set of upper bounds of ''A'', and let ''A''^l 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''^u)^l = ''A'';
it is ordered by inclusion. The Dedekind-MacNeille completion is generally a sublattice of the lattice of order ideals; ''S'' is embedded in it in the same way.
The Dedekind-MacNeille completion of a Boolean algebra is a complete Boolean algebra, whereas that of a distributive lattice need not be a distributive or even modular lattice.

Another generalization: surreal numbers

A construction similar to Dedekind cuts is used for the construction of surreal numbers.

Allusions

In his chapter on Henri Bergson, the author C.E.M. Joad employed imagery that was similar to Dedekind's concept of the cut. Joad was trying to explain how Bergson saw the mind as an instrument that projected permanent objects onto the experience of constant change. "The intellect, then, is a purely practical faculty, which has evolved for the purposes of action. What it does is to take the ceaseless, living flow of which the universe is composed and to make cuts across it, inserting artificial stops or gaps in what is really a continuous and indivisible process. The effect of these stops or gaps is to produce the impression of a world of apparently solid objects. These have no existence as separate objects in reality; they are, as it were, the design or pattern which our intellects have impressed on reality to serve our purposes." This is reminiscent of Dedekind's creation of a new irrational number at every gap in the continuous number line at which there is no existing real number.^[1]

Bibliography

★ Dedekind, Richard, ''Essays on the Theory of Numbers'', "Continuity and Irrational Numbers," Dover: New York, ISBN 0-486-21010-3

References

1. Great Philosophies of the World, C.E.M. Joad, Ch. VI, "The Philosophy of Change," 1930:Jonathan Cape and Harrison Smith, Inc.

See also

★ Cauchy sequence