TITLE: Is there an "algebraic" way to construct the reals?
QUESTION [11 upvotes]: It's possible to construct $\mathbb{Q}$ from $\mathbb{Z}$ by constructing $\mathbb{Z}$'s field of fractions, and it's possible to construct $\mathbb{C}$ from $\mathbb{R}$ by adjoining $\sqrt{-1}$ to $\mathbb{R}$.
In both cases, the construction is done purely algebraically. I.e. we only rely on the operations of our given structure to build the new structure. But at no point do we have to rely on the order properties of $\mathbb{Z}$ or $\mathbb{R}$ to get to $\mathbb{Q}$ or $\mathbb{C}$.
Every construction of $\mathbb{R}$ that I'm familiar with ultimately comes down to endowing $\mathbb{Q}$ with its usual order, and then imposing the completeness axiom on it to recover the rest of the real numbers.
Is it possible to get to $\mathbb{R}$ from $\mathbb{Q}$ without relying on the ordering properties of $\mathbb{Q}$?
Alternatively (relatedly?): There is the notion of a greatest common divisor for an arbitrary ring. This notion doesn't rely on any ordering properties; just algebraic ones. Is it possible to recover an order relation on $\mathbb{Q}$ using the GCD relation on $\mathbb{Z}$, then to impose completeness on $\mathbb{Q}$ and obtain $\mathbb{R}$, and then subsequently re-cast completeness in some algebraic manner? Thus defining $\mathbb{R}$ in purely algebraic terms?

REPLY [9 votes]: Noah Schweber has already explained that $\mathbb R$ cannot be obtained from $\mathbb Q$ or $\mathbb Z$ by any purely algebraic process or even by any first-order definable process.  Any construction of $\mathbb R$ must involve some second-order notions like arbitrary subsets or arbitrary infinite sequences (from $\mathbb Q$ or $\mathbb Z$).
There is, however, a construction of the reals that, though still using arbitrary sets (as it must for the reasons in Noah's answer) may look more algebraic and may be more to your liking. It's due to Steve Schanuel, who called it the Eudoxus reals, and you can read about it and find further references at https://ncatlab.org/nlab/show/Eudoxus+real+number# .
One nice property of this construction is that it starts with $\mathbb Z$ and does not first construct $\mathbb Q$ on the way to $\mathbb R$. Another is that it defines multiplication in a way that isn't obviously commutative (though commutativity isn't very hard to prove).