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Construction of $\mathbb{R}$ — The Axiom of Completeness (Part 1)

MATH 4630 · Chapter 1: The Real Numbers

The real numbers $\mathbb{R}$ are fundamentally one of the important objects for real analysis. Although we use real numbers constantly, it is not immediately clear which of their properties should be taken as assumptions and which should be proved. In this handout, we will describe the real number system axiomatically. Rather than constructing $\mathbb{R}$ from scratch, we will list a small number of fundamental properties, called axioms—that characterize the real numbers and distinguish them from other familiar number systems.

Axiom 1: Field Axioms

Let $\mathbb{R}$ be a set. We assume there exist two binary operations on $\mathbb{R}$, called $+$ (addition) and $\cdot$ (multiplication), such that $(\mathbb{R},+,\cdot)$ is a . This means that for all $x,y,z\in\mathbb{R}$:

Closure: $x+y\in\mathbb{R}$ and $x\cdot y\in\mathbb{R}$.
Associativity: $(x+y)+z = x+(y+z)$, $(x\cdot y)\cdot z = x\cdot(y\cdot z)$.
Commutativity: $x+y=y+x$, $x\cdot y=y\cdot x$.
Identities: There exist $0,1\in\mathbb{R}$ such that $x+0=x$ and $x\cdot 1=x$.
Inverses:
- (Additive inverse) For every $x$, there exists $-x$ such that $x+(-x)=0$.
- (Multiplicative inverse) For every $x\neq 0$, there exists $x^{-1}$ such that $x\cdot x^{-1}=1$.
Distributive Law: $x(y+z)=xy+xz$.

We define subtraction and division by:

$$x-y = x + (-y) \qquad\text{and}\qquad x\div y = x \cdot y^{-1} \quad (y\neq 0).$$

Fact: For each $x\in\mathbb{R}$, the additive inverse is unique, and if $x\neq 0$, the multiplicative inverse is unique.

The "Wish List"

The following statements about real numbers are all true, but they are not explicitly listed in Axiom 1. Some will follow from Axiom 1, while others require additional axioms.

For all $x\in\mathbb{R}$, $x\cdot 0=0$.
For all $x\in\mathbb{R}$, $-x = (-1)\cdot x$.
For all nonzero real numbers $a$, $a\neq -a$.
$\mathbb{R}$ is an infinite set.
There exists a real number $x$ such that $x^2=2$.

Exercise

Prove that for all $x\in\mathbb{R}$, $x\cdot 0=0$ using only the field axioms.

Axiom 2: Order Axioms

There exists a subset $\mathbb{R}^{+}\subseteq\mathbb{R}$, called the set of . This set satisfies:

If $x,y\in\mathbb{R}^{+}$, then $x+y\in$ and $x\cdot y\in$.
For every real number $a$, exactly one of the following is true:
- $a$ is .
- $a$ is .
- $-a$ is .

We call elements of $\mathbb{R}^{+}$ . The set of real numbers is the complement of $\mathbb{R}^{+}\cup\{0\}$ in $\mathbb{R}$, and we denote it by $\mathbb{R}^{-}$.

Exercise

Verify the following facts using Axiom 2.

The real number $1$ is positive.
There exists a negative real number.
For all nonzero real numbers $a$, $a\neq -a$.

Note

The rational numbers $\mathbb{Q}$ satisfy both Axiom 1 and Axiom 2.

So What is Missing?

If $\mathbb{Q}$ already satisfies Axioms 1 and 2, why do we need $\mathbb{R}$?

Thought Experiment

There is no rational number whose square is $2$.

So we can't hope that every positive real number has a square root using only Axiom 1 and Axiom 2. We need one more axiom—an axiom that will "fill the holes" left by rational numbers:

Axiom 3: The Axiom of Completeness

Every subset of $\mathbb{R}$ that is bounded above has a .

We will treat this as an : something we may use without proof.

In the next handout, we will unpack what "bounded above" and "least upper bound" mean, and we will learn how to compute $\sup A$ and $\inf A$ in examples.

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