Abstract:
Conservativity spectrum is a weakly descending sequence of ordinals ordn(T) characterizing the strength of a given arithmetical theory T w.r.t. provability of reflection principles of complexity Πn, for each n>0. We study a generalization of this notion to the language with transfinitely many Tarskian truth definitions introduced in [1] and to stronger reflection principles corresponding to the classes of hyperarithmetical hierarchy. We establish a correspondence of conservativity spectra and points of a generalized Ignatied model introduced by D. Fernandez-Duque and J. Joosten [2]. This gives an explicit characterization of the set of sequences of ordinals corresponding to conservativity spectra. We also show that the results of [1] easily yield the so-called Schmerl formulas for iterated reflection principles of predicative strength that are a basic tool for establishing various proof-theortic results on predicative theories.