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Autostability spectra for decidable structures

Published online by Cambridge University Press:  14 October 2016


NIKOLAY BAZHENOV
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia Email: bazhenov@math.nsc.ru
Corresponding
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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 above 0 (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 above 0 (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 to n-constructivizations.


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Copyright © Cambridge University Press 2016 

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