Mathematical Logic Quarterly 47 (1):51-66 (2001)

Abstract
A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left computable iff it is the supremum of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable sequences of rational numbers we introduce a non-collapsing hierarchy {Σn, Πn, Δn : n ∈ ℕ} of real numbers. We characterize the classes Σ2, Π2 and Δ2 in various ways and give several interesting examples
Keywords Computable real number  Recursively approximable real number  Arithmetical hierarchy
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DOI 10.1002/1521-3870(200101)47:1
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Relative Randomness and Real Closed Fields.Alexander Raichev - 2005 - Journal of Symbolic Logic 70 (1):319 - 330.

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