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COMPUTABLE POLISH GROUP ACTIONS

Published online by Cambridge University Press:  01 August 2018

ALEXANDER MELNIKOV  and
ANTONIO MONTALBÁN
ALEXANDER MELNIKOV
Affiliation:
THE INSTITUTE OF NATURAL AND MATHEMATICAL SCIENCES MASSEY UNIVERSITY AUCKLAND, NEW ZEALANDE-mail:alexander.g.melnikov@gmail.com
ANTONIO MONTALBÁN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA, USAE-mail:antonio@math.berkeley.eduURL: www.math.berkeley.edu/∼antonio
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Abstract

Using methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$-action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

Keywords

03D9903C5703D45computable analysiscomputable algebradescriptive set theorytopological group actions
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 443 - 460
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

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