I've always intuitively understood this set in intuitive sense, as "all numbers on the number line". However, now I want to know the formal definition:
Consider the set of rational numbers, $\mathbb{Q}$.
For any two Cauchy sequences of rational numbers $X=⟨x_n⟩,Y=⟨y_n⟩$, define an equivalence relation between the two as:
$X≡Y⟺∀ϵ>0:∃n∈ \mathbb{N}:∀i,j>n:|x_i−y_j|<ϵ$
The real numbers are the set of all equivalence classes $[[⟨x_n⟩]]$ of Cauchy sequences of rational numbers.
Most importantly, how do we deduce the axioms of the real number field from this definition?