  In mathematical logic there are two seemingly distinct kinds of principles
called "reflection principles." Semantic reflection principles assert that if a
formula holds in the whole universe, then it holds in a set-sized model.
Syntactic reflection principles assert that every provable sentence from some
complexity class is true. In this paper we study connections between these two
kinds of reflection principles in the setting of second-order arithmetic. We
prove that, for a large swathe of theories, $\omega$-model reflection is
equivalent to the claim that arbitrary iterations of uniform $\Pi^1_1$
reflection along countable well-orderings are $\Pi^1_1$-sound. This result
yields uniform ordinal analyses of theories with strength between
$\mathsf{ACA}_0$ and $\mathsf{ATR}$. The main technical novelty of our analysis
is the introduction of the notion of the proof-theoretic dilator of a theory
$T$, which is the operator on countable ordinals that maps the order-type of
$\prec$ to the proof-theoretic ordinal of $T+\mathsf{WO}(\prec)$. We obtain
precise results about the growth of proof-theoretic dilators as a function of
provable $\omega$-model reflection. This approach enables us to simultaneously
obtain not only $\Pi^0_1$, $\Pi^0_2$, and $\Pi^1_1$ ordinals but also
reverse-mathematical theorems for well-ordering principles.
