Reverse plane geometry, anyone? I refer to Greenberg's wonderful 2010 MAA article "Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries". There, and in his book, Greenberg defines a Hilbert plane as a model of the 13 Hilbert's first-order axioms of plane geometry. 

How many elementary equivalence classes of Hilbert planes are there? 

Recall that two Hilbert planes are elementarily equivalent if they satisfy the same first-order sentences.
A rough estimate from scanning through Greenberg's is that there are at least $10$ elementary equivalence classes. For example, among Euclidean planes, Greenberg already lists two other elementary equivalence classes: namely semi-Euclidean planes (where the angles in each triangle sum to two right angles) and Pythagorean planes (in which right-angled satisfy the Pythagorean identity). 
Sorting out this zoo of plane geometries would profit from the joint effort of many mathematicians. Polymath, maybe?
 A: There are exactly $2^{\aleph_0}$ elementary equivalence classes of Hilbert planes.
Let $P$ be a subset of the set of odd primes.  Let $K_P$ be the smallest field extension of $\mathbb{Q}(\{2^{1/p}:p \in P\})$ in $\mathbb{R}$ that is closed under taking square roots of positive elements, and let $H_P$ be the plane $K_P^2$.  For each odd prime $p$, there is a first-order sentence $S_p$ expressing the existence of a sequence of $p+1$ line segments whose lengths form a geometric progression such that the last segment is twice as long as the first; it holds in $K_P^2$ if and only if $2^{1/p} \in K_P$, which by degree considerations is equivalent to $p \in P$.  Thus for subsets $P$ and $Q$ of the set of odd primes, $H_P$ and $H_Q$ are elementarily equivalent if and only if $P=Q$.  The number of choices for $P$ is $2^{\aleph_0}$.
On the other hand, there are only countably many first-order sentences, and the number of elementary equivalence classes is at most the number of subsets of the set of first-order sentences, so there are at most $2^{\aleph_0}$ elementary equivalence classes.
