Cutting Through the Confusion: How Dedekind Cuts Build the Real Numbers

When we learn about numbers in school, we typically start with the natural numbers (1, 2, 3, …), move on to the integers (…, -2, -1, 0, 1, 2, …), and then to the rational numbers (numbers that can be expressed as fractions). We then draw a real line and assume that every point on the line exists and is called a real number, which then contains rationals, irrationals, integers, natural numbers and so on.

But how do we construct these real numbers?

Before we dive into the construction, let’s first understand one of the main concepts that would be used here — completeness.

Completeness

Since the time of ancient mathematicians, we know that the existing numbers on a real line cannot be all expressed by rational numbers. That the line contains certain, ‘gaps’, ‘holes’ or ‘jumps’, that cannot be filled by rational numbers. These are the irrational numbers.

An intuitive idea of the completeness concept is: All gaps on a real line which is covered by a real number — rational or irrational — means that the real line is complete.

So, if the goal is to construct the real numbers then, the goal is to show the completeness of real numbers.

How?

The completeness axiom states that every nonempty subset of the real numbers that is bounded above has a least upper bound. In other words, if we have a set of real numbers that has an upper bound (i.e., a number greater than or equal to all the numbers in the set), then there exists a smallest number that is greater than or equal to all the numbers in the set, but, less than or equal to the upper bound. This number is called the least upper bound or the supremum of the set. (Although the least upper bound does not necessarily belongs to the set.)

Mathematically, one needs to verify the two properties of a least upper bound or supremum. An s ∈ R is a least upper bound for a set A ⊆ R if

(i) s is an upper bound for A, and

(ii) if b is an upper bound for A, then s ≤ b.

So, the basic idea is to show that the set has a least upper bound when the given set is bounded above.

The above property of least upper bound to show completeness is not satisfied by the set of rational numbers. This why we need the entire set of real numbers to satisfy completeness, not just the rationals.

Dedekind cuts

There are various equivalent ways of formally describing the completeness property. For instance, one common notion of completeness, as mentioned previously, is the least upper bound property or the supremum property. Depending on how the problem of “filling the gaps” is approached, different but equivalent notions of completeness arise. In this article, one of the classical approaches will be demonstrated in detail: Dedekind’s construction through cuts on Q.

Dedekind cuts is one way to define the real numbers. But what exactly is a Dedekind cut? It’s a way of partitioning the rational numbers into two sets based on a certain property. Specifically, a Dedekind cut is a partition of the rational numbers into two sets A and B such that:

1. A and B are non-empty
2. A and B are disjoint (i.e., they have no elements in common)
3. Every element of A is less than every element of B
4. A has no largest element

The last property is important, as it distinguishes Dedekind cuts from other ways of partitioning the rationals. If A had a largest element, we could simply call that element the cut and be done with it. But because A has no largest element, we need to use the cut itself to define a new number.

We can think of a Dedekind cut as representing a point on the number line, rational or irrational. The set A represents all the rational numbers less than that point, while the set B represents all the rational numbers greater than or equal to that point. For example, we could define the irrational number √2 as the Dedekind cut (A, B), where A is the set of all rational numbers less than √2 and B is the set of all rational numbers greater than or equal to √2.

So, each Dedekind cut corresponds to a unique real number, and conversely, each real number can be represented by a Dedekind cut. We define the set of real numbers R as the set of all Dedekind cuts on Q.

There are many other ways to construct real numbers. The set of real numbers R constructed by Dedekind cuts is referred to as the Dedekind real numbers. The set of all Dedekind cuts form a complete ordered field.

What does this mean?

To understand this, we need to define some terms.

A field is a mathematical structure consisting of a set of elements along with two operations, usually called addition and multiplication, that satisfy certain axioms.

An ordered field is a field that has a total order relation that is compatible with the field operations. It contains a subset of elements closed under addition and multiplication and having the property that every element in the field is either 0, in the subset, or has its additive inverse in the subset.

A complete ordered field is an ordered field that satisfies the completeness axiom, which states that every non-empty subset of the field that is bounded above has a least upper bound.

The real numbers are an example of a complete ordered field.

We say that,

The complete ordered field of real numbers R is unique up to isomorphism.

This statement asserts that any two complete ordered fields that satisfy the same axioms as the real numbers are isomorphic to each other, meaning that there exists a one-to-one correspondence between the elements of the two fields that preserves the field operations and the order relation.

The proof of this statement is a non-trivial result in mathematical logic and model theory, and it involves showing that any two models of the same set of axioms are elementarily equivalent, meaning that they satisfy the same first-order sentences. This is known as the Löwenheim-Skolem theorem

Thus, this statement means that any two complete ordered fields that satisfy the same set of axioms as the real numbers are isomorphic to each other. In other words, any two such fields are essentially the same, up to a relabeling of the elements.

So, there is a unique isomorphism from one complete ordered field to another.

Now coming back to,

Set of all Dedekind cuts can be treated as a complete ordered field

Now, we need to show that the set of all Dedekind cuts can be treated as a complete ordered field, which is what we want the real numbers to be.

Why?

By defining the real numbers in terms of Dedekind cuts, we can prove that the set of real numbers is a complete ordered field, meaning that it satisfies the following properties:

1. It is a field, meaning that it has two binary operations (addition and multiplication) that satisfy certain axioms.
2. It is ordered, meaning that there is a total order relation that is compatible with the field operations.
3. It is complete, meaning that every non-empty subset that is bounded above has a least upper bound.

The completeness property is particularly important because it allows us to prove the existence of certain limits and to solve certain equations that would be impossible to solve otherwise. For example, the completeness property allows us to prove the intermediate value theorem, which states that if a continuous function takes on two different values at two points in its domain, then it must take on every value between those two points.

Thus, by showing that the set of all Dedekind cuts can be treated as a complete ordered field, we are able to define the real numbers in a rigorous and consistent way, and we are able to prove important properties such as the completeness property, which allows us to solve certain equations and prove important theorems in analysis.

So, we need to show that the set of all Dedekind cuts can form a complete ordered field.

We won’t go into the details of this proof here, but it involves showing that the set of Dedekind cuts satisfies all the properties we expect of the real numbers, such as the existence of additive and multiplicative inverses, the completeness property (every non-empty set with an upper bound has a least upper bound), and so on.

For detailed proofs refer to: [1] and [4].

Constructing the Real Numbers

Through the definitions, propositions and proofs given in [1] and [4], it has been shown that:

1. Dedekind real numbers are closed under addition and multiplication and hence showing that it is a field.
2. The Dedekind real numbers is an ordered field.
3. There is the important additional property of completeness. Thus, showing that the Dedekind real number system is Dedekind complete. [The Dedekind real number system has the supremum property.]

We previously stated that if the goal is to construct the real numbers then, the goal is to show the completeness of real numbers.

Seems like we have reached our goal.

In page 322 of [11], Dedekind himself gives a notion of completeness of the “straight line” (meaning the real number line) which is closely related to Dedekind cuts:

If all points on 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 dissection of the line into two pieces.

Notably, this notion of completeness uses Dedekind cuts on the real numbers and expresses a one-to-one correspondence between those cuts and the real numbers. In the proofs given in [1] and [4], this version of completeness has been formalised and expressed for general ordered fields.

Hence, this finishes one of the many possible ways of construction of the complete ordered field of real numbers.

Another classical approach leading to a different version of the completeness axiom, and hence constructing the real numbers, is the Cantor’s construction through Cauchy sequences which I have obviously not covered here, but, is equally interesting. [12]

One interesting consequence of the Dedekind cut construction is that it allows us to define irrational numbers as cuts of rational numbers, rather than as limits of sequences of rational numbers (which is how we usually define them in calculus). This may seem like a technicality, but it has important implications for the way we think about the real numbers and their properties.

In conclusion, Dedekind cuts provide a powerful tool for constructing the real numbers from the rational numbers. They allow us to represent every point on the number line as a cut of the rationals, and to define irrational numbers in a new and elegant way. While the construction may seem abstract and difficult to grasp at first, it is a fundamental concept in modern mathematics that underpins much of our understanding of the real numbers and their properties.

References

[1] BE Mathematical Extended Essay, Constructions of the real numbers a set theoretical approach by Lothar Sebastian Krapp, University of Oxford.

[2] Brigham Young University, Department of Mathematics, Math 341 Lecture #3 §1.3: The Axiom of Completeness.

[3] Construction of the real numbers, Wikipedia.

[4] Herbert B. Enderton, Elements of set theory, Academic Press, London, 1977.

[5] Why does the dedekind cut work well enough to define the reals?, Mathematics Stack Exchange.

[6] Dedekind cut, Wikipedia.

[7] Dedekind Cuts Handout by Rich Schwartz, Department of Mathematics, Brown University, 2014.

[8] Construction of the real numbers, Department of Mathematics, University of Washington.

[9] Completeness of the real numbers, Wikipedia.

[10] The Completeness Axiom for the Real Numbers, Introduction to Mathematical Analysis I (Lafferriere, Lafferriere, and Nguyen), Chapter 1: Tools for Analysis, Section 1.5, LibreTexts Mathematics.

[11] Richard Dedekind, Gesammelte mathematische Werke, vol. 3, 1931, reprint, Chelsea Publishing Co., New York, 1969.

[12] Lecture #1, 18.095 Lecture Series in Mathematics, Department of Mathematics, MIT, IAP 2015.