The infinitary logic $L_{\omega_1, \omega}$ extends first-order logic by allowing countable disjunctions and conjunctions of formulas. Every countable structure can be described up to isomorphism (within the class of countable structures) by an $L_{\omega_1, \omega}$ sentence. This gives rise to a particular way of measuring the complexity of countable structures: there is a natural alternation hierarchy $(\Pi^{in}_{\alpha}: \alpha < \omega_1)$ of $L_{\omega_1, \omega}$ formulas, and the Scott rank of a structure A is the smallest ordinal $\alpha$ such that A can be described up to isomorphism by a $\Pi^{in}_{\alpha+1}$ sentence.
In recent years, beginning with a paper by Montalbán and Rossegger, the Scott rank of models of arithmetic has attracted some attention. We now know, for instance, that every nonstandard pointwise definable model of PA has Scott rank at least $\omega$, that all other nonstandard models of PA must have rank at least $\omega+1$, and that recursively saturated models of PA have rank exactly $\omega+1$. Since proper subtheories of PA are of interest in many areas of logic (including reverse mathematics), it is also quite natural to ask about possible Scott ranks of models that satisfy some fragment of PA but not all of it. In particular, we would like to understand the Scott spectrum (i.e. the set of possible Scott ranks of models) of the theories $B\Sigma_n + \neg I\Sigma_n$ and $I\Sigma_n + \neg B\Sigma_{n+1}$, where $B\Sigma_n$ is the usual collection (a.k.a. bounding) principle for $\Sigma_n$ formulas.
We prove that every nonstandard model of $B\Sigma_n$ must have Scott rank at least n+1, and that this lower bound is tight: in fact, it is realized by the most familiar kinds of models of $B\Sigma_n + \neg I\Sigma_n$ and of $I\Sigma_n + \neg B\Sigma_{n+1}$ dating back to the work of Paris and Kirby. Moreover, we are able to give an almost exact characterization of the Scott spectrum of $I\Sigma_n + \neg B\Sigma_{n+1}$ (what remains is to determine whether one specific finite ordinal belongs to the spectrum) and to obtain less precise but also quite detailed information about the Scott spectrum of $B\Sigma_n + \neg I\Sigma_n$. Techniques that turn out to be useful in this area include the characterization of collection as a pigeonhole principle, automorphisms of models of collection, partial recursive saturation, and the "very low basis theorem" of Hájek-Kučera.
This is joint work in progress with Mateusz Łełyk and Patryk Szlufik.