Axiomatic introduction to the Real Numbers

I am sure that you probably know, intuitively, about the Real Numbers. However, this resource will disregard intuition of the Real Numbers, and will give a formal, Axiomatic approach to the Real Numbers.

## Contents

* 1 The Rationals
* 1.1 On Order
* 2 Limitations of the rational numbers
* 2.1 On the Least-upper-bound property
* 2.1.1 Definitions

## The Rationals[edit | edit source]

We start off from the rationals, as later we use them to define the Real Numbers. We can define the rationals as

Q = { p q : p , q ∈ Z , q ≠ 0 } {\displaystyle \mathbb {Q} =\\{{\frac {p}{q}}:p,q\in \mathbb {Z} ,q\neq 0\\}} . We can add, subtract, multiply and divide (except by 0) in the usual way. We call  Q {\displaystyle \mathbb {Q} } a field.

### On Order[edit | edit source]

I add to this in which the rational numbers are also an ordered field. That is, a field with an order  ≤ {\displaystyle \leq } such that  a ≤ b -> a + c ≤ b + c {\displaystyle a\leq b\rightarrow a+c\leq b+c} , and also  0 ≤ a {\displaystyle 0\leq a} and  0 ≤ b {\displaystyle 0\leq b} implies  0 ≤ a b {\displaystyle 0\leq ab} .

However, why do we need the real numbers at all, if we can use them just fine? The next section will elaborate on the limitations of the rational numbers.

## Limitations of the rational numbers[edit | edit source]

### On the Least-upper-bound property[edit | edit source]

#### Definitions[edit | edit source]

We go over the property of Least-upper-bound first. We start by defining the "prerequisite definitions": Say if we have a subset  S {\displaystyle S} of an ordered field. It is bounded above if there exists a number  b {\displaystyle b} in said ordered field s.t. ∀ x ∈ S , b ≥ x {\displaystyle \forall x\in S,b\geq x} .  b {\displaystyle b} is known as an upper bound of  S {\displaystyle S} .

A supremum β {\displaystyle \beta } of  S {\displaystyle S} satisfies 2 properties: It is an upper bound of  S {\displaystyle S} , and all other upper bounds of S are greater than or equal to  β {\displaystyle \beta } . In other words, a supremum of a set is the least upper bound of a set. As an example, take the set  { x ∈ N : x − 1 , x ≠ 0 } {\displaystyle \\{x\in \mathbb {N} :x^{-1},x\neq 0\\}} , in which we denote by  F {\displaystyle F} . It is bounded above, as there exists a number (say  3 2 {\displaystyle {\frac {3}{2}}} ) for which it is greater for all elements in  F {\displaystyle F} , and  3 2 {\displaystyle {\frac {3}{2}}} is an upper bound. The supremum of  F {\displaystyle F} is 1.

Also, we look back at  S {\displaystyle S} . It is bounded below if there exists a number  c {\displaystyle c} in said ordered field s.t. ∀ x ∈ S , c ≤ x {\displaystyle \forall x\in S,c\leq x} , and  c {\displaystyle c} is a lower bound. The infimum, or greatest lower bound of  S {\displaystyle S} is a lower bound of  S {\displaystyle S} that is greater that all other lower bounds of  S {\displaystyle S} .

The property of least-upper-bound then states that any subset  X ∈ Y {\displaystyle X\in Y} (whatever set  Y {\displaystyle Y} may be) that is bounded above also has a supremum. The rational numbers, though all subsets of the rational numbers are indeed bounded above, it is not always the case every subset in  Q {\displaystyle \mathbb {Q} } has a supremum.

Theorem.  Q {\displaystyle \mathbb {Q} } does NOT have the property of Least-upper-bound.

Proof. We give the example of the subset  { x ∈ Q : x 2 ≤ 2 } {\displaystyle \\{x\in \mathbb {Q} :x^{2}\leq 2\\}} . This subset has an upper bound, with the example of  x > 1.5 {\displaystyle x>1.5} . However, it does NOT have a supremum, as  2 ∉ Q {\displaystyle {\sqrt {2}}\not \in \mathbb {Q} } , even being bounded above. Therefore,  Q {\displaystyle \mathbb {Q} } does NOT have the property of Least-upper-bound.  ◻ {\displaystyle \Box }