Characterizing the continuum using only the notion of midpoint Is it possible to categorically characterize the continuum using only the notion of midpoint? I assume some kind of Dedekind-cut-axiom will be necessary, but I don't know how to define order (see my related question).
 A: Let me start with some formalism. 
Definition. A midpoint algebra (also known as a medial algebra) is a set $A$ with a binary "midpoint" operation $(x,y)\mapsto x|y$ satisfying the following axioms:


*

*$x|x=x$.

*$x|y=y|x$.

*$(u|v)|(x|y) = (u|x)|(v|y)$.  

*Horn cancelation law: $x|y = x|z  =>  y = z$. (This law is not always required, but I will assume it.) 
It is a good exercise to check that every convex subset of every vector space over ${\mathbb Q}$ equipped with the midpoint operation
$$
x|y= \frac{1}{2}(x+y)
$$
is a midpoint algebra. For instance, every subinterval in ${\mathbb R}$ will have this property. 
Also, every torsion-free diadic abelian group equipped with the midpoint operation 
$$
x|y= \frac{1}{2}(x+y)
$$
satisfies theses axioms. (For lack of a better name, I say that an abelian group $A$ is diadic if for each $a\in A$  there exists $b\in A$ such that $b+b=a$.) For instance, every subinterval in the additive group of diadic rational numbers satisfies axioms of a midpoint algebra. 
In order to eliminate possibility of intervals, we assume the "doubling axiom": 


*For  each pair $x, y\in A$ there exists $z=d(x,y)\in A$ such that $y=x|z$. The cancelation law implies that such $z$ is unique. 


Assume now that $A$ is a nonempty midpoint algebra satisfying axioms 1-5. Fix (at random) an element $e\in A$ and define the following binary operation:
$$
a+b:= d(0, a|b).
$$ 
If $A$ is a diadic abelian group with the neutral element $e$, it is clear that $a+b$ is the original addition operation on $A$. For a general midpoint algebra satisfying Axioms 1-5, you check that $(A, +)$ is a diadic abelian group. 
However, the midpoint operation on ${\mathbb R}$ (and, more generally, on every divisible abelian group) satisfies higher divisibility properties. For instance, using divisibility by 3, one obtains (for any pair $a, b\in A$) existence of $x, y$ such that $x=a|y, y=x|b$. There are several options how to proceed from here. For instance, you can add countably many divisibility axioms for $A$ (division by each $n$, $n\in {\mathbb N}$). Assuming this set of axioms, you obtain that the abelian group $(A,+)$ defined above is torsion free and divisible. Such a group has a natural structure of a vector space over ${\mathbb Q}$ and, hence, isomorphic to a direct sum
$$
\oplus_{j\in J} {\mathbb Q}. 
$$
If the index set $J$ has cardinality of continuum, then such $A$ is isomorphic to $({\mathbb R},+)$ which, therefore, yields a characterization of ${\mathbb R}$ as a midpoint algebra, as a divisible midpoint algebra of cardinality of continuum satisfying axioms 1-5. 
If you do not like having the divisibility axioms, you can proceed via an order: You postulate that the median algebra has a compatible total order (equivalently, the associated divisible abelian group is orderable). This is obviously expressible as a 2nd order sentence. Now, assume, in addition, that the total order is complete, i.e. every nonempty bounded (above) subset admits a least upper bound. (This is again a 2nd order sentence.) Lastly, Holder's theorem implies that every completely orderable abelian group is isomorphic to ${\mathbb R}$. This gives another characterization of   ${\mathbb R}$ as a midpoint algebra. 
