How can you prove the real number properties (I believe there are about 16 of them.) using only Zermelo-Fraenkel Set Theory with the Axiom of Choice? I am a high school math teacher and a graduate student studying real analysis.  My professor charged us with the task of finding such proofs.  I have exhausted Google.  I find only lists of these properties because they are widely accepted as axioms, which require no proof.
 A: There are many possible constructions of $\mathbb{R}$ using the standard axioms, and all that remains is to prove that these constructions satisfy the appropriate properties.
Essentially, $\mathbb{R}$ is defined as the completion of $\mathbb{Q}$ with respect to its usual distance - i.e. the unique, totally ordered field containing $\mathbb{Q}$ such that every Cauchy sequence converges. This means that all gaps in the rational numbers will be filled.
Formally, a Cauchy sequence is a sequence $(a_n)$ of rational numbers such that as $n,m\in\mathbb{N}$ become arbitrarily high, $a_n-a_m$ becomes arbitrary close to zero. We say that two Cauchy sequences $(a_n)$, $(b_n)$ are equivalent if $a_n-b_n$ converges to zero as $n\rightarrow\infty$. It is easily checked that this is an equivalence relation, and that two convergent sequences are equivalent if and only if they converge to the same limit.
We now define $\mathbb{R}$ as the set of all equivalence classes of Cauchy sequences under this relation, where $\mathbb{Q}$ is embedded into $\mathbb{R}$ by identifying each $q\in\mathbb{Q}$ with the sequence $q_n=q$ for all $n$. Elements of $\mathbb{R}$ can be added, subtracted, multiplied and divided by performing these operations on the sequences elementwise. A total ordering can be placed on $\mathbb{R}$ via $a\leq b$ if $a_n\leq b_n$ for sufficiently high $n$.
It only remains to check that these operations are well defined, and that they satisfy all the appropriate properties.
The idea behind this construction of $\mathbb{R}$ is to identify real numbers with any sequence of rational numbers converging to them. To see this construction in greater generality, try Googling 'the completion of a metric space', 'fields with an absolute value', or 'completion of a topological field'.
