Conservativity spectra and generalized Ignatiev model

TL;DR

This paper extends the concept of conservativity spectra to languages with transfinite truth definitions, linking them to generalized Ignatiev models and establishing key formulas for iterated reflection principles.

Contribution

It introduces a generalized framework for conservativity spectra in transfinite languages and connects them to new Ignatiev models, advancing understanding of reflection principles.

Findings

Established correspondence between conservativity spectra and generalized Ignatiev model points.

Proved Schmerl formulas for iterated reflection principles of predicative strength.

Extended the theory of conservativity spectra to transfinite truth languages.

Abstract

We study a generalization of the notion of conservativity spectrum of an arithmetical theory to a language with transfinitely many truth definitions. We establish a correspondence of conservativity spectra and points of a generalized Ignatied model introduced and studied by D. Fern\'andez-Duque and J. Joosten. We also prove the so-called Schmerl formulas for iterated reflection principles of predicative strength.