Download pdf
Proof of the Nonexistence of an Algorithm That Tells If a Point Is on a Closed Set’s Boundary or Exterior
- 1 Valley Christian High School, San Jose, 95111, CA, USA
- 2 University of Leeds, Leeds, LS2 9JT, UK
* Author to whom correspondence should be addressed.
Abstract
This study investigates the connection between the undecidability of certain sets and the problem of determining the position of a point relative to the boundary or exterior of a closed set. We introduce a new algorithm, referred to as Algorithm A, which generates a closed set at regular intervals. By employing a reductio ad absurdum argument, we demonstrate that no algorithm exists that can consistently determine whether a point lies on the boundary or in the exterior of a closed set. This finding highlights the limitations of algorithmic approaches in the context of constructive mathematics.
Keywords
Algorithm, undecidable, closed set
[1]. E. Bishop, D. Bridges: Constructive analysis Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 279 Springer-Verlag, Berlin (1985)[1,5]
[2]. B. A. Kushner: Lectures on constructive mathematical analysis (in Russian) Monographs in Mathematical Logic and Foundations of Mathematics. Iz- dat. “Nauka”, Moscow, 1973. 447 pp., English translation in Translations of Mathematical Monographs, 60 American Mathematical Society, Providence. [2]
[3]. Shen, A., and Nikolai Konstantinovich Vereshchagin. Computable Functions. American Mathematical Society, 2003. [3]
[4]. Kelly, John L. General Topology John L. Kelly. Springer, 1975. [4]
[5]. A. M. Turing: Corrections, Proc. Lond. Math. Soc., ser. 2, 43 (1937), 544-546 [5]
[6]. E. Bishop, D. Bridges: Constructive analysis Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 279 Springer-Verlag, Berlin (1985)
[7]. N. A. Sanin (Shanin): Constructive real numbers and constructive functional spaces (in Russian) Trudy Mat. Inst. Steklov 67 (1962) 15–294. English translation in Amer. Math. Soc., Providence R.I. (1968) [6]
[8]. N. Vereshchagin & A. Shen: Algorithms and randomness. In Proceedings of the 14th Annual IEEE Conference on Computational Complexity (1999), pp. 12–21. IEEE. [7]
Cite this article
Fan,J.;Li,J. (2025). Proof of the Nonexistence of an Algorithm That Tells If a Point Is on a Closed Set’s Boundary or Exterior. Theoretical and Natural Science,108,88-91.
Data availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
Disclaimer/Publisher's Note
The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
About volume
Volume title: Proceedings of the 4th International Conference on Computing Innovation and Applied Physics
© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution (CC BY) license. Authors who
publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons
Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this
series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published
version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial
publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and
during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See
Open access policy for details).