Reducing $\omega$-model reflection to iterated syntactic reflection

Abstract

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.