Set theoretic construction of real numbers

Continued from previous blog. Same reference: A Second course in analysis by M Ram Murty, Hindustan Book Agency.

Section 1.3

The rational number is insufficient to “measure” all the lengths that arise in the “real world.” This was discovered by the ancient school of Pythagoras which viewed the world through a strange mix of mathematics and mysticism. The aphorism “all is number” seems to have been the underlying mantra of the Pythagoreans. In our modern digital world, this mantra imposes the universality more than in any earlier age.

Pythagoras seems to have been a contemporaryof the Buddha in India, of Confucius and Lao-Tzu in China. He seems to have travelled widely in Egypt, Babylon and India. The Pythagoreans believed in reincarnation and this was wedded to their strict adherence to vegetarianism for by eating meat they might be eating a friend!They are credited to the discovery of both the words philosophy and mathematics. The word “philosophy” literally means the love of wisdom and the word “mathematics” means “that which is learned.” Philosophy then, for the Pythagoreans was seen as a byproduct of mathematics.

The celebrated theorem of Pythagoras dealing with the length of the hypotenuse of a right angled triangle was known to earlier cultures stretching far back in time to Babylon, China and India. It is possible that Pythagoras learned of this theorem during his extensive peregrinations. But the Pythagoreans could see the mysticism inherent in the theorem, for the rule applied to all right angled triangles and there was an element of universality in it and in mathematics in general.

But when the Pythagoreans declared “all is number” they were referring to whole numbers and it is accurate to say that the concept of number had not been clearly articulated at that time. With the underlying idea that whole numbers govern the universe, their theory of ratio and proportion was in for a rude shock. Legend has it that the Pythagorean who discovered that \sqrt{2} is an irrational number was drowned !

The irrationality of \sqrt{2} represents a major episode in the history of mathematics and we owe this to the unfortunate Pythagorean who lost his life by discovering it. The proof is by contradiction and invokes the rudimentary arithmetical idea of a prime number, a concept first enunciated in Euclid’s Elements. A natural number is called a prime number if it has no proper divisors. Thus, the sequence of prime numbers begins as 2,3,5,7,11, \ldots (The number one is not included in the list of prime numberss. It it called a unit, a concept which generalizes to rings.) Two natural numbers a and b are said to be coprime if they have no common prime factor. When we write a fraction a/b in “lowest termss”, we are writing it such that a and b are coprime. We sometimes write (a,b)=1 to indicate that a and b are coprime. This is not to be confused with the concept of an ordered pair introduced earlier.

The irrationality of \sqrt{2} is established by using the method of contradiction. It is instructive to see all the details. If \sqrt{2} is a rational number, then we can write

\sqrt{2}=\frac{a}{b} such that (a,b)=1

Upon squaring both sides, we see

2b^{2}=a^{2}

which implies that a is even since the left hand side of the equation is patently an even number. Writing a=2c, we now get

b^{2}=2c^{2}

and so b is even. But this contradicts the coprimality of a and b.

This was the theorem that led to the demise of the heroic Pythagorean for it violated the aphorism “all is number” because the ancients viewed numbers as only whole numbers. They also allowed for rational numbers since they can be represented as ratio and proportion of whole quantities. This innocuous theorem opens the door for the discovery of real numbers. For it signals the lack of the “least upper bound property” in the realm of rational numbers.

The student should keep in mind that so far, we have only constructed the universe of rational numbers and thus any further definitions must be given only in terms of rational numbers. A set A of rational numbers is said to be bounded if there is a rational number M such that |x| \leq M for all x \in A. An upper bound for A is any rational number u such that x \leq u for all x \in A. A lower bound for A is a rational number l such that l \leq x for all x \in A. A rational number v is called a least upper bound for A if v \leq u for every upper bound u of A. A rational number m is called a greatest lower bound for A if l \leq m for every lower bound l of A. Thus, a set A is bounded if and only if it has both an upper bound and a lower bound.

Several questions now arise: given a set A of rational numbers, does it always have a least upper bound? Does it have a greatest lower bound? If so, are they unique?

The theorem discovered by the Pythagoreans can be explained as follows. Nascent in this example are many of the fundamental properties we would expect of the real numbers. Essential to the understanding is the familiar fact that the sequence of rational numbers \frac{1}{n} gets smaller and smaller and tends to zero.

If we consider the set

A = \{ x \in \mathcal{Q}: x^{2} \leq 2\}

then A has no least upper bound. Indeed if u is a least upper bound, x \leq u for any x \in A. By definition, u is a rational number and so cannot first equal \sqrt{2}. If u < \sqrt{2}, then we can choose a natural number N such that

\frac{1}{N} < \sqrt{2}-u

so that u + \frac{1}{N}< \sqrt{2}

contradicting that u is an upper bound for the set A. If u>\sqrt{2}, then we can similarly find N such that

u - \frac{1}{N}> \sqrt{2},

which again contradicts that u is the least upper bound for A. Thus, A does not have a least upper bound in the realm of rational numbers.

Digress: We digress a bit here from Dr. M Ram Murty’s above text and reproduce a nice “little proof”‘ from Rudin’s Analysis text illustrating that the rational number line has gaps or holes in it..

****

As we know there exists no rational solution to p^{2}=2. We now examine this situation a little more closely. Let A be the set of all positive rationals p such that p^{2}<2 and let B consist of all positive rationals p such that p^{2}>2. We shall show that A contains no largest rational number and B contains no smallest rational number.

More explicitly, for every p \in A we can find a rational q in A such that p<q, and for every p in B, we can find a rational q in B such that q < p.

To do this, we associate with each rational number p>0 the number q such that:

q = p - \frac{p^{2}-2}{p+2}=\frac{2p+1}{p+2}…call this I.

Thus, q^{2}-2 = \frac{2(p^{2}-2)}{(p+2)^{2}}….call this II.

If p is in A, then p^{2}-2<0, (1) above shows that q >p, and II shows that q^{2}<2. Thus, q is in A.

If p is in B, then p^{2}-2>0, (I) shows that 0<q<p, and II shows that q^{2}>2. Thus, q is in B.

Conclusion: the purpose of the above discussion is to show that the rational number system has gaps, in spite of the fact that in between any two rational numbers, there is another: if r, s \in \mathcal{Q}, and if r<s, then r < \frac{r+s}{2}<s. The real number system fills these gaps. This is the principle reason for the fundamental role which it plays in analysis.

***

Two ways of constructing the real numbers were proposed independently by Richard Dedekind (1831-1916) and Augustin Cauchy (1789-1857). Each method has its virtues. The method of Dedekind, using what are called Dedekind cuts is closer to the axiomatic foundations we have been discussing and is suggested by the example discussing \sqrt{2} above. The method of Cauchy using what are now called Cauchy sequences has a wider applicability. In our example above, an essential role is played by the ordering of the rational numbers. The method of Dedekind cuts uses this idea in a fundamental way. It coincides with our intuitive and visual idea of the real numbers as being “points on the number line stretching from minus infinity to plus infinity.”

A Dedekind cut in Q is a partition of Q into two sets A and B with the following properties:

  1. A and B are both non-empty.
  2. A  \bigcup B = \mathcal{Q}
  3. A is “closed downwards”, that is, if r \in A and q<r, then q \in A;
  4. B is “closed upwards”, that is, if q \in B and q<r, then r \in B.
  5. A contains no greatest element, that is, there is no q \in A such that r \leq q for all r \in A.

The set R of real numbers is then defined as the set of all Dedekind cuts.

For example, the irrational number \sqrt{2} is defined by the Dedekind cut:

\mathcal{Q} = A \bigcup B, and A = \{ x \in \mathcal{Q}: x^{2}<2\}, and B = \mathcal{Q}-A.

For example, the number zero is represented by A \bigcup B with A being the set of all non-zero negative rational numbers and B being the set of all non-zero positive rational numbers. As the Dedekind cut A \bigcup B is uniquely defined by A, with B being the complement of A in the set of rationals, we may as well associate A to each real number, or even think of A as the real number. One would then have to define how one would work with this definition from the perspective of addition, multiplication and so on. This is not difficult to do. For instance, we have

A_{1}< A_{2} \Longleftrightarrow A_{1} \subset A_{2}, and A_{1} \leq A_{2} \subseteq A_{2}.

Addition is simple enough:

A_{1}+ A_{2} = \{ a_{1}+a_{2}: a_{1} \in A_{1}, a_{2} \in A_{2}\}

Multiplication is defined (rather cumbersomely) by setting A_{1}.A_{2} to be

\{ a_{1}.a_{2}: a_{1} \in A_{1}, a_{2} \in A_{2}, a_{1}, a_{2} \geq 0\} \bigcup \{ q \in Q: q \leq 0\} if A_{1}, A_{2} \geq 0

-(A_{1}.(-A_{2})), if A_{1} \geq 0, and A_{2}<0

-((-A_{1}).A_{2}), if A_{1}<0, A_{2} \geq 0

(-A_{1}).(-A_{2}), if A_{1},A_{2}<0

The absolute value function is then

|A| = A, if A>0; |A|=0, if A=0; |A|=-A, if A <0.

One can prove that all the usual operations with numbers satisfy the expected properties of commutativity, associativity, and distributivity.

By contrast, the method of Cauchy sequences begins with a definition of convergence. In some ways, it is motivated by our discussion of \sqrt{2} earlier. If A is the Dedekind cut representing \sqrt{2} , then the least upper bound of A “is” \sqrt{2}. Thus, the number \sqrt{2} can be thought of as a limit of a sequence of rational numbers. But there can be many such sequences and so one needs a more formal approach. The underlying observation is that a limit of a sequence of rational numbers need not be rational.

This is perhaps best illustrated by Euler’s series for e. Students of calculus are familiar with

e = \sum_{j=0}^{\infty}\frac{1}{j!}

and the right hand side can be thought of as a limit of a sequence of rational numbers

S_{n}=\sum_{j=0}^{n}\frac{1}{j!}

The irrationality of e is easily proved by the method of contradiction. Suppose that e=a/b for two coprime natural numbers a and b. Then,

b!e = \sum_{j=0}^{b}\frac{b!}{j!} + \sum_{j=b+1}^{\infty}\frac{b!}{j!} …call this 1.1

and the second term on the right hand side is strictly less than the geometric series

\sum_{k=1}^{\infty} \frac{1}{(b+1)^{k}}=\frac{1}{b}

which is less than or equal to 1. But the left hand side of 1.1 is a positive integer, the first term on the right hand side of 1.1 is a positive integer and the second term is not an integer, which is a contradiction.

What this example shows is that the “limit” of the sequence of partial sums S_{n} (each of which is a rational number) is the irrational number e.

The reader is familiar with the usual notion of convergence. We say a sequence of real numbers x_{n} converges to L if given \epsilon>0 there exists N such that

|x_{n}-L|< \epsilon for all n \geq N

One may want to view real numbers as “limits of sequences of rational numbers.”But in our formal construction of numbers, several difficulties arise with this definition. The first is that as we have so far only constructed the rational numbers, we need to take the x_{n}‘s to be all rational numbers. Secondly, the limit L need not be a rational number as the example with e shows. This motivates the formal definition of a Cauchy sequence of rational numbers.

A sequence q_{n} in \mathcal{Q} is called a Cauchy sequence if given \epsilon>0, there is an N such that

|q_{n}-q_{m}|< \epsilon for all m,n \geq N

Now it is easy to see that any convergent sequence is a Cauchy sequence. For reference, we record this below:

Theorem 1.2: Any convergent sequence q_{n} is a Cauchy sequence.

Proof of 1.2: Suppose that q_{n} converges to L. Then, choosing N such that

|q_{n}-L|< \frac{\epsilon}{2} for all n \geq N

we have for all m,n\geq N, by the triangle inequality

|q_{n}-q_{m}| \leq |q_{m}-L|+|L-q_{n}|<\epsilon.

QED.

Later, we will show that every Cauchy sequence converges. Thus, a sequence is Cauchy if and only if it is convergent and the two notions are equivalent. But the advantage of the notion of a Cauchy sequence is that the limit value L is not mentioned in the definition.

The construction of real numbers is now brought about by defining an equivalence relation on the set of all Cauchy sequences of rational numbers. Since we have constructed rational numbers, we are now allowed to construct the set of all Cauchy sequences of rational numbers. On this set, we say two Cauchy sequences \{ q_{n}\} and \{ r_{n} \} are equivalent if for any \epsilon>0, there exists a natural number N such that

|q_{n}-r_{n}|< \epsilon for all n \geq N.

This is easily seen to define an equivalence relation on the set of all Cauchy sequences of rational numbers. The set of real numbers \Re is then defined as the set of all equivalence classes. We will denote the equivalence class of the sequence \{ q_{n}\} by |\{ q_{n}\}|.

We leave as an exercise to verify that if \{ q_{n}\} and \{ r_{n}\} are two Cauchy sequences, then so are \{ q_{n}+r_{n}\} and \{ q_{n}.r_{n}\}. Thus, the usual operations of addition and multiplication are easily defined as:

[\{ q_{n}\}]+[\{ r_{n}\}] = [\{q_{n}+r_{n} \}]

[\{ q_{n}\}].[\{ r_{n}\}] = [\{ q_{n}.r_{n}\}]

One needs to verify that these are well-defined by checking that the definition does not depend on the choice of representative of the equivalence class.

The order relation of the real numbers can also be defined. Thus, viewing real numbers as equivalence classes or Cauchy sequences of rational numbers, we define [\{ q_{n}\}] < [\{ r_{n}\}] if there is an N such that q_{n}<r_{n} for all n \geq N. We also write [\{ q_{n}\}] \leq [\{ r_{n}\}] if either [\{ q_{n}\}] < [\{ r_{n}\}] or [\{ q_{n}\}] = [\{ r_{n}\}]. Again, these order relations have the expected properties.

Finally, the absolute value function can also be defined. We set

$latex|[\{ q_{n}\}]| = [\{ |q_{n}|\}]$

and the reader can easily verify that indeed the right hand side is a Cauchy sequence if \{ q_{n}\} is Cauchy.

The absolute value has the usual properties, the most notable being the triangle inequality

|[\{ q_{n}\}] + [\{ r_{n}\}]| \leq |[\{ q_{n}\}]|+|[\{ r_{n}\}]|

Having given this formal definition of the real numbers using Cauchy sequences, it is convenient to drop the sequence notation and just represent the real numbers as single letters. Thus, if a, b are real numbers, the triangle inequality is the familiar: |a+b| \leq |a|+|b|

It is now convenient to visualize the set of real numbers in the usual way as points of the line stretching from -\infty to +\infty.

By the very construction, the set of real numbers has the property of completeness in that every Cauchy sequence converges. The reader will also observe that the construction depended only on the property of the absolute value viewed as a metric to measure the distance between two rational numbers. This signals a wider applicability of the method of Cauchy sequences valid more generally in metric spaces which we discuss later.

With the construction of the real numbers, we have restored the least upper bound property and can now define that a sequence of numbers (rational or real) x_{n} converges to a number L \in \Re if given any \epsilon >0, there is an N such that

|x_{n}-L|<\epsilon for all n \geq N.

A non-empty subset A of \Re is said to be bounded below if there is a number s \in \Re such that s \leq a for all a \in A. Any such number s is called a lower bound for A. Analogously, A is said to be bounded above if there is an upper bound for A. The set A is said to be bounded if it is both bounded below and bounded above and unbounded if it is not bounded.

The construction of the real numbers using the Dedekind cuts shows that every non-empty subset of \Re that is bounded above has a least upper bound denoted by \sup {A}. The least upper bound is also called the supremum. Every non-empty subset which is bounded below has a greatest lower bound denoted by \inf {A}. The greatest lower bound is also called the infimum. It is clear that if A and B are non-empty subsets of \Re such that A \subseteq B, then \sup{A} \leq  \sup{B}(see exercises). The following theorem gives a convenient characterization of the supremum.

Theorem 1.3 Let A be a non-empty subset of \Re that is bounded above. Then s=\sup {A} if and only if

(i) a \leq s for all a \in A, and

(ii) for any \epsilon>0, A \bigcap (s-\epsilon, s]\neq \phi.

Proof :

Let s=\sup {A}. Then s is an upper bound for A so that (i) holds. Now if (ii) were false, there would be an \epsilon>0 such that A \bigcap (s-\epsilon, s] = \phi. But then s-\epsilon would also be an upper bound for A, a contradiction. Conversely, suppose thatt (i) and (ii) hold. Then (i) implies s is an upper bound for A. If s were not the least upper bound for A, there would bea t <ssuch that a \leq t for alla \in A. Taking \epsilon = \frac{s-t}{2}>0 in (ii) we get that A \bigcap (s-\epsilon, s] \neq \phi. But s-\epsilon= t+\epsilonso that A \bigcap (t+\epsilon, s] \neq \phi which is a contradiction since a \leq t for all a \in A.

QED.

Theorem 1.3 can also be stated as follows. If A is a non-empty subset of \Re that is bounded above ands=\sup{A}, then either s \in A or A \bigcap (s-\epsilon, s] \neq \phi for all \epsilon >0. A similar theorem can be written for the infimum of a set.

Cheers,

Nalin Pithwa

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