## Journal of Symbolic Logic

### The approximation structure of a computably approximable real

George Barmpalias

#### Abstract

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.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 3 (2003), 885- 922.

Dates
First available in Project Euclid: 17 July 2003

https://projecteuclid.org/euclid.jsl/1058448447

Digital Object Identifier
doi:10.2178/jsl/1058448447

Mathematical Reviews number (MathSciNet)
MR2004d:03132

Zentralblatt MATH identifier
1056.03040

#### Citation

Barmpalias, George. The approximation structure of a computably approximable real. J. Symbolic Logic 68 (2003), no. 3, 885-- 922. doi:10.2178/jsl/1058448447. https://projecteuclid.org/euclid.jsl/1058448447

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