# Construction of The Real Numbers

DEFINITION1: The set $\mathbb{R}$ with at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of $\mathbb{R}$ is called a real number:

The function $+:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined as $\left( x,y \right)\to x+y\in \mathbb{R}$ for each $\left( x,y \right)$ in $\mathbb{R}\times \mathbb{R}$ satisfies the following properties:
I${{}_{1}}.\,\forall a,b\in \mathbb{R},a+b=b+a$,
I${{}_{2}}.\,\forall a,b,c\in \mathbb{R},a+(b+c)=(a+b)+c$,