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Primitive recursive real numbers
Author(s) -
Chen Qingliang,
Su Kaile,
Zheng Xizhong
Publication year - 2007
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/malq.200710005
Subject(s) - real number , mathematics , representation (politics) , dedekind cut , computable number , hierarchy , discrete mathematics , algebra over a field , pure mathematics , computable function , computable analysis , politics , political science , economics , law , market economy
In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind‐complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” (p. r., for short), these definitions lead to a number of different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b ‐adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)