Autostability spectra for decidable structures

Abstract

We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure$\mathcal{S}$, the SC-autostability spectrum of$\mathcal{S}$is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of$\mathcal{S}$. The degree of SC-autostability for$\mathcal{S}$is the least degree in the spectrum (if such a degree exists).We prove that for a computable successor ordinal α, every Turing degree c.e. in and above0(α)is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above0(2β+1)is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative ton-constructivizations.