Cantor’s Construction of the Real Numbers

By: Jonathan Rodriguez and Shivani Patel

Introduction:

In this class, we often work with different parts of the real numbers, such as natural numbers, integers, and rational numbers. We can easily take real numbers for granted since they seem to encompass everything that we work with in the class. In order to fully appreciate the real numbers, it is valuable to understand how we can “create” the real numbers. If we task ourselves with making all of the real numbers, it appears to be a daunting task. However, Cantor made the keen observation that the real numbers have a sequential form and thus created his construction of the reals. We can utilize our knowledge regarding Cauchy sequences to understand Cantor’s construction of R.

Before we get into the details of Cantor’s construction of $\mathbb R$ , we have some definitions and background information that is important to understand in the context of construction.

Rational Sequence: A sequence is function that has domain N, so a rational sequence is a function $\mathbb{N} \rightarrow \mathbb{Q}$ that results in an infinite list of numbers. It can also be written as $n \rightarrow s_n$ .For example, (0, 0.1, 0.1, 0.3, 0.4, 0.5,…) is a rational sequence.

Cauchy Sequence: A sequence $(a_n)$ is Cauchy if for any $\epsilon > 0$ , there exists some $N_1$ in the natural numbers such that $\mid$ $a_n - a_m$ $\mid$ $<$ $\epsilon$ for all $m,n \geq N_1$ . Colloquially, a Cauchy sequence is a sequence where the difference between two terms approaches 0.

Theorem: Every convergent rational sequence ( $(a_n) \rightarrow$ $q$ $\forall$ $q \in \mathbb Q$ ) is a Cauchy sequence.

Proof: By the definition of convergence, we know that for all arbitrary $\epsilon > 0$ there exist $n_0 \in N$ such that $\mid$ $a_m -q$ $\mid$ $<$ $\epsilon/2$ and $\mid$ $a_p -q$ $\mid$ $<$ $\epsilon/2$ and for all $m,p \geq n_0$ . Consider $\mid$ $a_m -a_p$ $\mid$ . By the triangle inequality, $\mid$ $a_m -a_p$ $\mid$ $=$ $\mid$ $a_m - a+a -a_p$ $\mid$ $\leq$ $\mid$ $a_m -a$ $\mid$ + $\mid$ $a -a_p$ $\mid$ . Further, $|a_m-a| + |a-a_p| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon$ . Therefore, $(a_n)$ is a Cauchy sequence.

The Cauchy Criterion builds upon this theorem.

Cauchy Criterion: A sequence converges if and only if it is a Cauchy sequence. In section 2.6 of our textbook, Abbott emphasizes the importance of the Cauchy Criterion in relation to real numbers. The Cauchy Criterion is proved by the Bolzano-Weierstrass Theorem which is proved by the Nested Interval Property. This chain of implications is important because it ultimately begins with the Axiom of Completeness, and because it outlines numerous fundamentals of the real numbers. Specifically, these theorems and properties combined suggest that the real numbers are complete and that they can be constructed. Furthermore, we see that the proof for the Cauchy Criterion only involves sequences that contain any real numbers. This relationship between these theorems and axioms is dependent on the completeness of the Real Numbers. When presented with an order field that is not complete, one will find that this relationship begins to break down since it is impossible to prove any of these theorems and axioms with only the field and order properties. This is the case for Cantor’s construction which consists of sequences that are comprised of rational numbers.

Equivalence Relation: Let S be a set of objects. A relation among pairs of elements of S is said to be an equivalence relation if the following three properties hold:

Reflexivity: for any s element of S, s is related to s.

Symmetry: for any s, t element of S, if s is related to t, then t is related to s.

Transitivity: for any s,t,r element of S, if s is related to t and t is related to r, then s is

related to r

Equivalence Classes: Let S be a set, with an equivalence relation on pairs of elements. For s element of S, the equivalence class , denoted by [s], is the set of all elements in S that are related to s. The for any s,t element of S, either [s]=[t] or [s] and [t] are disjoint.

Sequences of Rational Numbers:

In trying to construct the Real Numbers $\mathbb R$ , it is important to recognize that many times, real numbers are simply identified by an approximate rational number. Consider the real number =3.14169…. $\pi$ can be approximated by multiple rational number such as 3.14, 3.14159, or 3.1415926536. By continuing to list the infinite number of rational numbers that can be used to approximate the value of $\pi$ , a rational sequence is created whose terms get closer and closer to the value of $\pi$ .

In addition, a sequence of this nature would be a Cauchy sequence since the difference between terms is approaching 0. It is also important to realize that a Cauchy sequence is bounded, meaning that $\exists\ M$ such that $|a_n| \leq M\ \forall\ n \in \mathbb N$ . However, while a Cauchy sequence is bounded, one cannot assume that the sequence converges to a rational number. With this, Cantor considered the idea that sequences which do not converge to a rational number instead converge to a value known as an irrational number, as in the case of the rational sequence whose terms are approaching $\pi$ . Furthermore, by adding the numbers that Cauchy sequences “seem” to converge to, Cantor stated that the set of Rational Numbers $\mathbb Q$ would become complete.

This process of creating sequences of rational numbers that approximate real numbers leads to the important idea that a real number is simply a Cauchy sequence of rational approximations. While this idea is useful in constructing the Real Numbers, a problem arises when one takes into consideration the many sequences of rational numbers that can be used to approximate a real number. For example, $a_n =$ (3, 3,1, 3.14, 3.141, 3.1415, 3.14159, 3.1415926536, 3.1415926536, 3.1415926536…) and $b_n=$ (1, -1, 10, -10, 3, 3,1, 3.14, 3.141, 3.1415, 3.14159, 3.1415926536, 3.1415926536, 3.1415926536…) are two sequences that can be used to approximate the value of $\pi$ . In fact, there are infinitely many sequences that can be used to approximate $\pi$ , as well as every other real number. With this, Cantor’s solution was to divide rational sequences into equivalence classes.

Cauchy’s Construction of the Reals:

To begin, let $C_Q$ be the set of all rational Cauchy sequences. In order to eventually create equivalence classes, we begin with an equivalence relation in $C_Q$ . Define the relation as follows: For two sequences $(a_n)$ and $(b_n) \in \ C_Q$ , consider them to be equivalent if $(a_n - b_n \rightarrow 0)$ . At this point, it is important to note that this does not suggest that $a_n = b_n$ , instead it means that the “tails” of the sequences behave similarly. Also as a side note, we know that this is an equivalence relation because it is reflexive, symmetric, and transitive. Since we have established this equivalence relation, we can partition $C_Q$ into equivalence classes. These equivalence classes form the real numbers.

While it would be great if we could stop here, constructing the reals from the equivalence classes formed of rational Cauchy sequences requires that we ensure that this construction is an ordered field. Thus, one must begin by proving the field axioms, that is prove 1) equivalence classes are closed under addition and multiplication 2) addition and multiplication of equivalence classes is commutative, associative, and distributive 3) there exists additive and multiplicative identities 4) there exists additive and multiplicative inverses. Through a variety of proofs, one can see that Cantor’s construction is in fact a field. Next, one should prove that this field is ordered by proving the order axiom: 1) For any $x \in \mathbb R$ , exactly one of $x > 0,\ x = 0,\ 0 < x$ is true 2) If $x, y > 0$ then $x + y > 0$ and $xy > 0$ , 3) If $x > y$ then $x + z > y + z\ \forall \ z$ . Again, through a couple proofs, one can see that that Cantor’s construction is ordered. By having proofs of these axioms, one can see how the rational numbers are a subfield of the real numbers. In addition, one can also see the reasoning behind Cantor’s construction of the real numbers as the equivalence classes formed of rational Cauchy sequences.

Sources:

https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/readings-notes/MIT18_100BF10_Const_of_R.pdf

http://www.math.uni-konstanz.de/~krapp/Constructions_of_the_real_numbers.pdf

Jeremy LeCrone says:

September 23, 2017 at 8:00 pm

Awesome blog! Thank you for exploring this very interesting construction of the “bedrock” of our analysis class. I hope you found it interesting (even in the face of how “daunting” it is) to explore ways to actually confirm it is possible to create the ordered field we brought into existence “by assumption” in class.

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Math 320: Real Analysis

Cantor’s Construction of the Real Numbers

Introduction:

Sequences of Rational Numbers:

Cauchy’s Construction of the Reals:

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Introduction:

Sequences of Rational Numbers:

Cauchy’s Construction of the Reals:

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