# 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} }

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…

## Tables from this paper

## 20 Citations

A chain of inclusion relations in computable analysis

- Mathematics
- 1969

Introduction. Within the subfield of the reals, the computable numbers, it is possible to develop an analysis with constructive definitions for all the usual concepts: sequence, convergent sequence,…

Decidability and Undecidability in the Limit

- Mathematics
- 1988

In some problems of numerical analysis or optimization, we are faced with non-convergent sequences and have to obtain information from them (for examples see [23] , [24] , [26] , [30] and [37]).…

Computational Models For Feasible Real Analysis

- Computer Science
- 1990

This expository work discusses the conventional oracle Turing machine model of recursive analysis and proposes an alternative one based on uniform Boolean circuit families. This replacement model is…

Toward Abstract Numerical Analysis

- MathematicsJACM
- 1973

This paper deals with a technique for proving that certain problems of numerical analysis are numerically unsolvable, and the number of necessary function evaluations is taken as the measure of computational complexity.

The Concect of Effective Method Applied to Computational Problems of Linear Algebra

- MathematicsJ. Comput. Syst. Sci.
- 1971

The failure in computable analysis of a classical existence theorem for differential equations

- Mathematics
- 1971

An example is given of a uniformly continuous constructive function f(x, y) withf(0, 0) = 0, such that the differential equation y' =f(x, y) with the initial condition y(0) = 0 has no constructive…

Beyond the universal Turing machine

- Computer Science
- 1999

An emerging field is described, that of nonclassical computability and non classical computing machinery, and a philosophical defence of its foundations is provided.

## References

SHOWING 1-10 OF 31 REFERENCES

Recursive real numbers

- Mathematics
- 1954

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

- MathematicsJ. 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

- Mathematics, Computer ScienceJ. 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