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The Arithmetical Hierarchy of Real Numbers
Author(s) -
Zheng Xizhong,
Weihrauch Klaus
Publication year - 2001
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200101)47:1<51::aid-malq51>3.0.co;2-w
Subject(s) - infimum and supremum , mathematics , rational number , arithmetic function , real number , sequence (biology) , hierarchy , computable analysis , computable number , number theory , combinatorics , discrete mathematics , limit (mathematics) , rational point , limit of a sequence , computable function , algebraic number , mathematical analysis , genetics , economics , market economy , biology
A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left (right) computable iff 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 Σ 2 , Π 2 and Δ 2 in various ways and give several interesting examples.