Cantor's view of real numbers

In order to introduce the concept of real numbers we follow the ideas presented by Georg Cantor. First, we need to discuss equivalence relation defined on an abstract set X.

A binary relation on X that is reflexive, symmetric and transitive is called
equivalence relation.

In other words, the relation $\sim$ on X is called equivalence if the following three conditions are met.

(Reflexive relation) a ~ a
(Symmetric relation) if a ~ b then b ~ a
(Transitive relation) if a ~ b and b ~ c then a ~ c

Example. For a given fixed natural number p consider a relation on the set of integers Z; two integers n and m are called equivalent

$n \sim m$

$p \vert (n - m) .$

This is an equivalence relation on Z. One can see that under this relation the set Z is divided in p different classes. Each class consists of equivalent with each other integers. The set of such equivalence classes is called quotient set and denoted as

${\rm Z} / \sim$

The quotient set considered here is very important and in the literature often denoted as Z_p.

Now consider a sequence of rational numbers {A_n}. This sequence is said to have limit L if

$\forall \varepsilon > 0\;\;\exist \;\;M\in {\rm N} \;\;\mbox{ such that } \forall\;n > M\;\;\vert L - A_n \vert < \varepsilon$

Let GSQ denote the set of all possible Cauchy sequences of rational numbers. We call to sequences {A_n} and {B_n} from GSQ equivalent and write

{A_n} $\sim$ {B_n}

if, and only if,

$\lim_{n \to \infty } ( A_n - B_n ) = 0.$

According to Georg Cantor the corresponding quotient set

GSQ /~

is called real numbers and denoted as