Conservativity Spectra and Joosten–Fernández Model

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 natural correspondence of conservativity spectra and points of a special Kripke model introduced and studied by D. Fernández–Duque and J. Joosten. For iterated reflection principles over theories of truth definitions we also establish conservation results analogous to the well-known Schmerl formulas.