Generating real numbers How can I describe the positive set of real numbers in terms of algebraic structures?
For example, I can describe a set of integers $(\mathbb Z, +)$ as $\langle 1\rangle$.
 A: I would argue there is no such definition.
First, note that just by cardinality considerations (remember that the reals are uncountable) we can never build $\mathbb{R}$ by starting with some countable set (e.g. $\mathbb{Q}$) and closing off under countably many functions (this is a good exercise). I think this already rules out any algebraic characterization.
We can even do better: the Lowenheim-Skolem theorem implies that there is no way we can build the reals from (say) the rationals via first-order logic (which on the face of it is considerably stronger than just closing under some algebraic operations, although the proof of LS is really a reduction to this case). Again, this is the cardinality barrier at work.
I think the moral of this is:

In order to get an uncountable structure from a countable one, you need to go beyond pure algebra.

This may involve bringing in some (low level) set theory (e.g. looking at the powerset of an algebraic structure), or some topology (e.g. topological completions), or some analysis (e.g. closure under Cauchy sequences), and in fact all three of these are basically different repackagings of the same construction; but the point is that you have to do something new.

That said, "algebraic characterization" is a vague notion. Here's a characterization of the reals which you might accept as algebraic (I don't, but to each their own):

$\mathbb{R}$ is the largest Archimedean real closed field. That is, if $F$ is any Archimedean real closed field, then there is a (unique!) field homomorphism from $F$ to $\mathbb{R}$.

This definition is non-algebraic in two ways: "largest" (which involves quantifying over all possible fields!) and "Archimedean" (which is an infinitary first-order sentence). Of the two, I think "largest" is the significant offender (if you really get down to it, we need something like "Archimedean" to even pin down $\mathbb{N}$!). Note how uncharacteristic this is, from an algebraic point of view: most constructions (e.g. the algebraic closure of a field) are the smallest structure with [properties]. We can get around this by using a different characterization of $\mathbb{R}$:

$\mathbb{R}$ is (up to isomorphism) the unique complete ordered field.

However, the word "complete" there quantifies over all Cauchy sequences - again outside first-order logic. It does, however, get away from the "maximizing" picture of the previous definition.
