Reducing omega-model reflection to iterated syntactic reflection

Journal of Mathematical Logic 23 (2) (2021)
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Abstract

Journal of Mathematical Logic, Volume 23, Issue 02, August 2023. 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, [math]-model reflection is equivalent to the claim that arbitrary iterations of uniform [math] reflection along countable well-orderings are [math]-sound. This result yields uniform ordinal analyzes of theories with strength between [math] and [math]. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory [math], which is the operator on countable ordinals that maps the order-type of [math] to the proof-theoretic ordinal of [math]. We obtain precise results about the growth of proof-theoretic dilators as a function of provable [math]-model reflection. This approach enables us to simultaneously obtain not only [math], [math] and [math] ordinals but also reverse-mathematical theorems for well-ordering principles.

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James Walsh
New York University

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References found in this work

Transfinite recursive progressions of axiomatic theories.Solomon Feferman - 1962 - Journal of Symbolic Logic 27 (3):259-316.
Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems.Georg Kreisel & Azriel Lévy - 1968 - Zeitschrift für Mathematische Logic Und Grundlagen der Mathematik 14 (1):97--142.
Π12-logic, Part 1: Dilators.Jean-Yves Girard - 1981 - Annals of Mathematical Logic 21 (2):75-219.
Proof-theoretic analysis by iterated reflection.Lev D. Beklemishev - 2003 - Archive for Mathematical Logic 42 (6):515-552.

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