It is generally impossible to determine whether a rational point lies in the interior or on the boundary of a closed set algorithmically

Research Article
Open access

It is generally impossible to determine whether a rational point lies in the interior or on the boundary of a closed set algorithmically

Baiqing Liu 1 , Xinyao Zhu 2*
  • 1 Xi’an Jiaotong University    
  • 2 WLSA Shanghai Academy    
  • *corresponding author witty_zhu2008@outlook.com
Published on 26 December 2023 | https://doi.org/10.54254/2753-8818/28/20230399
TNS Vol.28
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-261-9
ISBN (Online): 978-1-83558-262-6

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

Liu,B.;Zhu,X. (2023). It is generally impossible to determine whether a rational point lies in the interior or on the boundary of a closed set algorithmically. Theoretical and Natural Science,28,151-155.
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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.

Cite this article

Liu,B.;Zhu,X. (2023). It is generally impossible to determine whether a rational point lies in the interior or on the boundary of a closed set algorithmically. Theoretical and Natural Science,28,151-155.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN：978-1-83558-261-9(Print) / 978-1-83558-262-6(Online)
Editor：Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.28
ISSN：2753-8818(Print) / 2753-8826(Online)

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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.

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