A language $L_{\omega_1\omega}$ generalizes an ordinary first-order language by allowing countably long disjunctions. If we take its nonlogical vocabulary to contain just a predicate for the successor relation, then every subset of $\mathbb N$ can be defined by some formula in the language.
How does the situation change if we consider the sublanguage which results from the extra requirement to form countable disjunctions only of r.e. sets of formulas?*
For example, is every $\Pi^1_1$ set definable?
Thankyou!
Max
*Note:
Let me try to clarify the extra requirement. (Please let me know if this doesn't work!)
Set up some reasonable coding of atomic formulas $A$ as $|A|$, of variables $v$ as $|v|$, and of pairs of integers $(a,b)$ as $\langle a,b\rangle$. Define a set of indices by induction as the smallest $I$ such that
$\langle 0,|A|\rangle\in I$ for every atomic $A$.
if $i\in I$, then $\langle 1,i\rangle\in I$
if $i\in I$, then $\langle 2,\langle|v|,i\rangle\rangle\in I$
if $W_e\subseteq I$, then $\langle 3,e\rangle\in I$.
Then obtain the corresponding subset of formulas of $L_{\omega_1\omega}$ in the natural way.