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DEGREES OF CATEGORICITY ON A CONE VIA η-SYSTEMS

Published online by Cambridge University Press:  21 March 2017

BARBARA F. CSIMA  and
MATTHEW HARRISON-TRAINOR
BARBARA F. CSIMA
Affiliation:
DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO 200 UNIVERSITY AVE. W. WATERLOO, ON N2L 3G1, CANADAE-mail: [email protected]: www.math.uwaterloo.ca/∼csima
MATTHEW HARRISON-TRAINOR
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USAE-mail: [email protected]: www.math.berkeley.edu/∼mattht
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Abstract

We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is ${\rm{\Delta }}_\alpha ^0 $-complete for some α. To prove this, we extend Montalbán’s η-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinal α and a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy ${\cal B}$ in the cone is a c.e. degree in ${\rm{\Delta }}_\alpha ^0\left( {\cal B} \right)$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions.

Keywords

computability theorycomputable structure theorydegrees of categoricitynatural structureon a cone
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 325 - 346
Copyright
Copyright © The Association for Symbolic Logic 2017 

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