Reducing ω-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, [Formula: see text]-model reflection is equivalent to the claim that arbitrary iterations of uniform [Formula: see text] reflection along countable well-orderings are [Formula: see text]-sound. This result yields uniform ordinal analyzes of theories with strength between [Formula: see text] and [Formula: see text]. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory [Formula: see text], which is the operator on countable ordinals that maps the order-type of [Formula: see text] to the proof-theoretic ordinal of [Formula: see text]. We obtain precise results about the growth of proof-theoretic dilators as a function of provable [Formula: see text]-model reflection. This approach enables us to simultaneously obtain not only [Formula: see text], [Formula: see text] and [Formula: see text] ordinals but also reverse-mathematical theorems for well-ordering principles.