Apostel's Axioms for the Real Numbers

There are many ways to construct the real numbers. I think being slightly outside the scope of Apostel 1, it takes a nonconstructive approach starting out rather far in the process:
* He takes the real numbers \mathbb{R} as undefined objects
* States a list of 10 axioms that the real numbers must satisfy
All properties of \mathbb{R} can be deduced from these ten axioms. I suppose that if we would consequently study a constructive approach, we could use all the results we find in Apostel by proving these ten axioms as theorems.
== Axioms ==
The ten axioms consist of: 6 field axioms, 3 order axioms, and a completeness axiom. Note that the six field axioms are the same as the definition of a field.