# DEGREES OF CATEGORICITY ON A CONE VIA η-SYSTEMS

@article{Csima2017DEGREESOC,
title={DEGREES OF CATEGORICITY ON A CONE VIA $\eta$-SYSTEMS},
author={Barbara F. Csima and Matthew Harrison-Trainor},
journal={The Journal of Symbolic Logic},
year={2017},
volume={82},
pages={325 - 346}
}
• Published 1 March 2017
• Mathematics
• The Journal of Symbolic Logic
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…
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## References

SHOWING 1-10 OF 38 REFERENCES
Degree Spectra of Relations on a Cone
This work represents the beginning of an investigation of the degree spectra of "natural" structures, and it is shown that any degree spectrum on a cone which strictly contains the $\Delta^0_2$ degrees must contain all of the 2-CEA degrees.
Degrees of categoricity of computable structures
• Mathematics, Computer Science
Arch. Math. Log.
• 2010
It is shown that for all n, degrees d.c.e. in and above 0(n) can be so realized, as can the degree 0(ω), which Turing degrees can be realized as degrees of categoricity.
Degrees That Are Not Degrees of Categoricity
• Mathematics, Computer Science
Notre Dame J. Formal Log.
• 2016
A large class of degrees that are not degrees of categoricity are demonstrated by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categricity.
Degrees coded in jumps of orderings
• J. Knight
• Mathematics
Journal of Symbolic Logic
• 1986
It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation.
Degrees of Categoricity and the Hyperarithmetic Hierarchy
• Mathematics, Computer Science
Notre Dame J. Formal Log.
• 2013
It is shown that for every computable ordinal α, 0 is the degree of categoricity of some computable structure A, and that for α a computable successor ordinal, every degree 2-c, i.e. in and above 0 is a degree ofategoricity.
RECURSIVE LABELLING SYSTEMS AND STABILITY OF RECURSIVE STRUCTURES IN HYPERARITHMETICAL DEGREES
We show that, under certain assumptions of recursiveness in 21, the recursive structure 21 is A^-stable for a < wfK if and only if there is an enumeration of 21 using a E^ set of recursive EQ
Categoricity Spectra for Rigid Structures
• Mathematics
Notre Dame J. Formal Log.
• 2016
This paper investigates spectra of categoricity for computable rigid structures with least degree and gives examples of rigid structures without degrees of categricity.
The axiom of determinateness and reduction principles in the analytical hierarchy
Let R be the set of all sets of natural numbers. A collection (ü of subsets of R satisfies a reduction principle if, for every A and 5 G d , there are A' and B'E® such that A'QA^B'QB, A'UB'=*AUB, and