(To clarify, my interest is mainly *lightface*, that is, $\Pi^1\_2$ instead of **$\bf \Pi^1\_2$**, although it doesn't particularly matter.)


This is just an idle curiosity. In logic, I find myself frequently working with the countable ordinals; this is a $\Pi^1\_1$ set of structures, in the sense that the set of reals coding well-orderings with domain $X\subseteq\omega$ is $\Pi^1\_1$. This is also a fairly natural class of structures, of interest outside logic as well; and other interesting classes of structures - e.g., torsion groups, Artinian rings, infinite graphs into which a given infinite graph $G$ does not embed - are also $\Pi^1\_1$ (and usually are vastly simpler than $\Pi^1\_1$).


Obviously, there are more complicated classes of structures; however, all the examples I know are pretty artificial. So my question is: what are some natural classes of countable structures, which are worse than $\Pi^1\_1$? 


In particular:



> 
> Is there a natural class of countable structures which is $\Pi^1\_2$ complete?
> 
> 
> 


(Of course, what "natural" means is subjective. What I mean by natural here is "of roughly equal interest to non-logicians as well-orderings," but examples from within logic would be okay if there is an interesting argument to be made that they will someday be of interest outside logic.)



Natural $\Pi^1_2$ (or worse) classes of structures?
An example that comes to mind is the set of recursive dilators; this is a $\Pi^1\_2$-complete set (Theorem 4.1 in Girard 1985, [Introduction to $\Pi^1\_2$-logic](http://dx.doi.org/10.1007/BF00486046)). Dilators have some use outside of logic (e.g.: Some uses of dilators in combinatorial problems [I](http://dx.doi.org/10.1090/conm/065/891241), [II](http://dx.doi.org/10.2307/2274952), [III](http://dx.doi.org/10.1007/BF01620619)).


**Added October 27:**


*Definition:* A dilator is functor from ON to ON (the large poset of ordinals and strictly increasing functions) preserving filtered colimits and pullbacks. A dilator is *weakly finite* if it maps finite ordinals to finite ordinals. A dilator is *recursive* if it is weakly finite and the morphism part is a recursive function from N to N under a standard coding of finite sequences as numbers.


Thus, weakly finite dilators are coded by reals, while recursive dilators are coded by numbers, and in the latter case forming a $\Pi^1\_2$-complete set, just as the usual set of codes of recursive ordinals forms a $\Pi^1\_1$-complete set.


