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The approximation structure of a computably approximable real. (English) Zbl 1056.03040

Summary: A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the \(0'\)-computable reals. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real \(x\) we assign a degree structure, the structure of all possible ways available to approximate \(x\). So the main criterion for such classification is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results.

MSC:

03F60 Constructive and recursive analysis
03D30 Other degrees and reducibilities in computability and recursion theory
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References:

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