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Nonexistence of Algorithm Deciding if the Fundamental Group of an Arbitrary Constructive Compact Set is Trivial
- 1 Dalian No.24 High School Cambridge International Education Department
- 2 College of Art and Science, New York University, New York
- 3 Nanjing Foreign Language School, Nanjing
- 4 Department of Mathematics, University of California, Berkeley
- 5 No.2 High School of East China Normal University
* Author to whom correspondence should be addressed.
Abstract
In this paper, we give an introduction to the compact space under Constructive Mathematics, ultimately showing that there exist no computer programs that can always determine if a fundamental group of a constructive compact topological space is trivial or not. We will begin with the non-halting case of the programs, which shows that the fundamental group of our selected set is trivial. From here, we will explore the halting case of the programs, which can be solved by arbitrarily picking a nontrivial subset of our selected set. The main result of this section is that we can show that the algorithm is decidable due to our initial construction, while realizing its contradiction with our input. We want to show that such an algorithm doesn’t exist by employing the proof-by-contradiction method.
Keywords
compact space, constructive mathematics, fundamental group, halting problem
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Cite this article
Jin,E.;Li,Y.;Qian,X.;Yao,K.;Yu,S. (2025). Nonexistence of Algorithm Deciding if the Fundamental Group of an Arbitrary Constructive Compact Set is Trivial. Theoretical and Natural Science,108,38-42.
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Volume title: Proceedings of the 4th International Conference on Computing Innovation and Applied Physics
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