TITLE: Natural examples of Reverse Mathematics outside classical analysis?
QUESTION [15 upvotes]: Harvey Friedman at the 1974 ICM motivated Reverse Mathematics by the
following statement:

When the theorem is proved from the right axioms, the axioms can be proved
from the theorem.

Reverse Mathematics has had many successes in finding the "right axioms,"
but to date mainly for theorems of classical analysis, where real numbers
(or equivalent infinite objects, such as sets of natural numbers)
are involved. This may be partly for historical reasons, since the subject
grew from the study of subsystems of second order number theory.
It seems to me that there are classical theorems in other fields that
also fit the Reverse Mathematics paradigm: add a strong axiom $A$ to a weak
theory $W$ and prove $A$ equivalent, over $W$, to a strong theorem $T$ of $W+A$.
The example I have in mind is where $W$ is the theory of a projective plane $P$,
which has three simple axioms about objects called "points" and "lines":


Any two points belong to a unique line.

Any two lines have a unique common point.

There exist at least four distinct points.



This $W$ is a weak theory with very few interesting theorems. But if we add to
$W$ the theorem of Pappus as an axiom $A$, then results of von
Staudt, Hilbert, and Hessenberg enable us to prove the theorem

$T$: The plane $P$ can be coordinatized by a field.

Conversely, if $T$ holds then we can prove $A$. This is because the Pappus axiom
$A$ states that certain points lie on a line, and $A$ can be proved once we have
coordinates in a field -- by computing the coordinates of the points in question
and showing that they satisfy a linear equation.
There are a few variations of this result. For example, if axiom $A$ is replaced
by the theorem of Desargues then the equivalent theorem $T$ of $W+A$ is that $P$
can be coordinatized by a skew field.
This leads me to the following questions. Are these results reasonable examples
of Reverse Mathematics? Are there other natural examples outside analysis?
Edit (Nov 15, 2014). Many thanks to the logicians who have
answered this question. It appears to me now that the term "reverse
mathematics" is too narrow for what I had in mind, but I am still
interested in examples of "finding the right axioms." An even more
elementary example than the Pappus axiom/theorem is the Euclid's
parallel axiom. It is needed to prove many theorems, e.g. Pythagorean
theorem, angle sum of a triangle equals $\pi$, ... and these theorems
also prove the axiom.

REPLY [9 votes]: I agree with Henry Towsner that the terminology "reverse mathematics" is generally used only for things in or near second-order arithmetic.  If, however, you're interested in situations where one tries to prove a theorem from some axioms and then show that the axiom actually follows from the theorem, regardless of whether these situations are called "reverse mathematics", then I'd point to the study of weak forms of the axiom of choice.  Here the base theory is ZF (or perhaps a variant ZFA that allows atoms), the additional axioms that one considers are the axiom of choice and weakenings of it, and one tries to show that various results of ordinary (not set-theoretic) mathematics are equivalent (over ZF) to one or another of these axioms.  The book "Consequences of the Axiom of Choice" by Paul Howard and Jean Rubin gives lots of examples.