Primitive recursive real numbers

Mathematical Logic Quarterly 53 (4‐5):365-380 (2007)
  Copy   BIBTEX

Abstract

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” , 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

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 91,202

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Primitive recursive real numbers.Qingliang Chen, Kaile Kaile & Xizhong Zheng - 2007 - Mathematical Logic Quarterly 53 (4):365-380.
Order‐free Recursion on the Real Numbers.Vasco Brattka - 1997 - Mathematical Logic Quarterly 43 (2):216-234.
H‐monotonically computable real numbers.Xizhong Zheng, Robert Rettinger & George Barmpalias - 2005 - Mathematical Logic Quarterly 51 (2):157-170.
Recursive Approximability of Real Numbers.Xizhong Zheng - 2002 - Mathematical Logic Quarterly 48 (S1):131-156.
Frege meets dedekind: A neologicist treatment of real analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
Non-constructive Properties of the Real Numbers.J. E. Rubin, K. Keremedis & Paul Howard - 2001 - Mathematical Logic Quarterly 47 (3):423-431.
The Arithmetical Hierarchy of Real Numbers.Xizhong Zheng & Klaus Weihrauch - 2001 - Mathematical Logic Quarterly 47 (1):51-66.
Real numbers and other completions.Fred Richman - 2008 - Mathematical Logic Quarterly 54 (1):98-108.

Analytics

Added to PP
2013-12-01

Downloads
30 (#502,094)

6 months
3 (#880,460)

Historical graph of downloads
How can I increase my downloads?

References found in this work

Nicht konstruktiv beweisbare sätze der analysis.Ernst Specker - 1949 - Journal of Symbolic Logic 14 (3):145-158.
Rekursive Funktionen.Raphael M. Robinson & Rozsa Peter - 1951 - Journal of Symbolic Logic 16 (4):280.
Criteria of constructibility for real numbers.John Myhill - 1953 - Journal of Symbolic Logic 18 (1):7-10.
The recursive irrationality of π.R. L. Goodstein - 1954 - Journal of Symbolic Logic 19 (4):267-274.
Recursive real numbers.A. H. Lachlan - 1963 - Journal of Symbolic Logic 28 (1):1-16.

Add more references