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:

I. AXIOMS OF ADDITION:

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,

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