Analysis in the Computable Number Field

@article{Aberth1968AnalysisIT,
  title={Analysis in the Computable Number Field},
  author={Oliver Aberth},
  journal={J. ACM},
  year={1968},
  volume={15},
  pages={275-299}
}
  • O. Aberth
  • Published 1 April 1968
  • Mathematics, Computer Science
  • J. ACM
It is well known that real variable analysis is nonconstructive. For example, although it is asserted that every bounded monotone sequence converges to a limit, there is no algorithm for obtaining this limit. This paper presents a constructive analysis which is restricted to a countable set of numbers, the field of computable numbers. These numbers are defined in a new way by employing the concept of “programmable functions.” The resultant analysis differs from real analysis in many important… 
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References

SHOWING 1-10 OF 31 REFERENCES
Recursive real numbers
TLDR
This work presents here some of the elementary properties of recursive real numbers, as defined below, which may be described intuitively as one for which the authors can effectively generate as long a decimal expansion as they wish.
Criteria of Constructibility for Real Numbers
  • J. Myhill
  • Mathematics, Computer Science
    J. Symb. Log.
  • 1953
The purpose of this paper is to prove two theorems and a conjecture (Conjecture II) announced in section 15 an earlier paper of the author's (cited as “ CT ”), and to compare them briefly with
Nicht Konstruktiv Beweisbare Satze Der Analysis
  • E. Specker
  • Mathematics, Computer Science
    J. Symb. Log.
  • 1949
Nach allgemeiner Ueberzeugung konnen gewisse Satze der Analysis nicht konstruktiv bewiesen werden.
Konstr2~ktive Analysis. Deutsche Verlag der Wissensehaften
  • Konstr2~ktive Analysis. Deutsche Verlag der Wissensehaften
  • 1961
Recursive Analysis. North-II
  • Recursive Analysis. North-II
  • 1961
Der Satz vom Maximum in der rekursive Analysis. Constructivity in Mathematics
  • Der Satz vom Maximum in der rekursive Analysis. Constructivity in Mathematics
  • 1959
On computable numbers , with au application to the Entseheidungs prob
  • Proc . Lond . Math . Soc .
  • 1959
Comp~tability and Un,~olvability. Me(ir,.tw-Hill
  • Comp~tability and Un,~olvability. Me(ir,.tw-Hill
  • 1958
On constructive functions
  • Trudy Mat. Inst. b,~. Steklova
  • 1958
...
1
2
3
4
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