Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 28, 26 December 2023
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It is generally impossible to determine whether a rational point lies in the interior or on the boundary of a closed set algorithmically
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Abstract
It seems true that we can almost always determine the position of a specific point when a set is given. Namely, we could assert whether this point lies at this set's interior, boundary, or exterior. However, this is not always the case in constructive mathematics. In this research, we will show that it is generally impossible to algorithmically determine whether a rational point lies in the interior of a closed productive set or on the boundary of it. We conduct our proof by making contradictions. Firstly, We used an unextendible algorithm to construct a rational point and a closed set on the natural line. Secondly, we reformulate the assumption "we could decide whether the point lies in the interior or on the boundary of a closed set” to “we could determine the program will eventually print 1". Thirdly, we constructed an extension of the program to all the positive integers, which is a contradiction to our assumption. Hence, we concluded that it is impossible to figure out the position of the rational point algorithmically.
Keywords
Unextendible Program, Constructive Mathematics, Closed Set, Rational Point
References
1. Bishop, E. Beeson, M. (2013). Foundations of constructive analysis. Ishi Press International.
2. Mints, G. E.: 1983, 'Stepwise Semantics of A. A. Markov' (Russian), a supplement to the Russian translation of the Handbook of Mathematical Logic, Nauka, Moscow, part IV, pp. 348-357.
3. Shanin N.A., “A hierarchy of ways of understanding judgments in constructive mathematics”, Problems of the constructive direction in mathematics. Part 6, Trudy Mat. Inst. Steklov., 129, 1973, 203–266; Proc. Steklov Inst. Math., 129 (1973), 209–271.
4. Kushner, B. A. (1985). Lectures on constructive mathematical analysis. American Mathematical Society.
5. Mandelkern, M. “Constructive Mathematics.” Mathematics Magazine, vol. 58, no. 5, 1985, pp. 272–80.
6. Turing, A. M. (1936). On Computable Numbers, with An Application to the Entscheidungs problem. Proceedings of the London Mathematical Society, 42, 1936.
7. Shen, A., and Vereshchagin N.K. Computable Functions. AMS Press, 2003.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation
- ISBN (Print)
- 978-1-83558-261-9
- ISBN (Online)
- 978-1-83558-262-6
- Published Date
- 26 December 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/28/20230399
- Copyright
- 26 December 2023
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited