The arithmetical hierarchy of real numbers

Abstract

A real number is computable if it is the limit of an effectively converging computable sequence of rational numbers, and left (right) computable if it is the supremum (infimum) of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable (multiple) sequences of rational numbers we introduce a non-collapsing hierarchy Σ n, Π n, Δ n: n ∈ ℕ of real numbers. We characterize the classes Σ2 Π 2 and Δ 2 in various ways and give several interesting examples.