We define the real number system to be a set \(\mathbb{R}\) together with an ordered pair of functions from \(\mathbb{R} \times \mathbb{R}\) into \(\mathbb{R}\) that satisfy the seven properties listed below.
The first five axioms are the field properties (because these axioms are satisfied by any field):

0. Closure: \(\forall a,b \in \mathbb{R}: (a+b),(a\cdot b) \in \mathbb{R}\)
1. Commutivity : \(a + b = b + a\) and \(a \cdot b = b \cdot a\)
2. Associativity : \((a + b) + c = a + (b + c)\) and \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
3. Distributivity : \(a \cdot (b + c) = a \cdot b + a \cdot c\)
4. Existence of natural elements: \(\exists 0 : a + 0 = a\) and \(\exists 1 : a \cdot 1 = a\) and \(0 \neq 1\)
5. Existence of inverses: \(\exists (-a) : a + (-a) = 0\) and for non-zero \(a\) \(\exists a^{-1} : a \cdot a^{-1} = 1\)

Then we have the ordered field axiom:

6. There is a subset \(\mathbb{R}_{+}\) of \(\mathbb{R}\) such that:
   • if \(a,b \in \mathbb{R}_{+}\) then \((a+b), (a\cdot b) \in \mathbb{R}_{+}\)
   • \(\forall a \in \mathbb{R}\), one and only of of the following is true:
     • \(a \in \mathbb{R}_{+}\)
     • \(a = 0\)
     • \(-a \in \mathbb{R}_{+}\)

Finally, we have the completeness axiom:

7. A nonempty set of real numbers that is bounded from above has a least upper bound.