## Journal of Symbolic Logic

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

### The approximation structure of a computably approximable real

#### 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

**Permanent link to this document**

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

**Subjects**

Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]

Secondary: 03D30: Other degrees and reducibilities

**Keywords**

Computably approximable reals approximation structure immunity properties

#### 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