The Non-universality of an Algorithm to decide if a point is in the exterior or on the boundary of a closed constructive set

Authors

  • Yicong Bao
  • Yiqian Li
  • Sicong Liu
  • Longjun Yuan

DOI:

https://doi.org/10.61173/9jx13362

Keywords:

Constructive mathematics, algorithm, ratio-nal point, close set

Abstract

It seems we can find if a point is on the boundary or in the exterior of a closed constructive set. However, this is not true in constructive math. This paper investigates the algorithmic decidability of determining the position of rational points relative to closed sets given as the intersection of closed intervals with rational endpoints.

References

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[3] Robinson, A., & Bishop, E. (1968). Foundations of constructive analysis. ˜the œAmerican Mathematical Monthly, 75(8), 920. https://doi.org/10.2307/2314383

[4] Constructive Mathematics | Internet Encyclopedia of Philosophy. (n.d.). https://iep.utm.edu/constructivemathematics/

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[6] Shen A. and Vereshchagin N. K. (2002) “Computable Functions”. Translated by V. N. Dubrovskii. American Mathematical Society. Existence of unextendible programs Theorem 10 in Shen book, existence of the enumerable but undecidable sets Theorem 11 in Shen book

[7] Turing A. M. , “On computable numbers, with an application to the Entscheidungsproblem” Proc. London Math. Soc. (2) , 42 Dean&Francis Yicong Bao, Yiqian Li, Sicong Liu, Longjun Yuan (1937) pp. 230–265

[8] Turing A. M. , “On computable numbers with an application to the Entscheidungsproblem, a correction” Proc. London Math. Soc. (2) , 43 (1937) pp. 544–546

[9] Tseitin G. S. , “Algorithmic operators in constructive metric spaces” Trudy Mat. Inst. Steklov., 67 (1962) pp. 295–361 (In Russian)

[10] Shanin N. S. , “Constructive real numbers and constructive function spaces” Trudy Mat. Inst. Steklov. , 67 (1962) pp. 15– 294 (In Russian)

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Published

2025-07-06

Issue

Vol. 3 No. 3 (2025): Volume 3, Issue 3

Section

Articles

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