Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 25, 20 December 2023
Open
Access | Article
The axiom of choice and its applications: An exploration of the '100 prisoners problem'
* Author to whom correspondence should be addressed.
Abstract
The Axiom of Choice (AC), a cardinal principle in set theory, postulates that for any assortment of disjoint non-empty sets, it’s possible to construct a new set by selecting one element from each set in the collection. Using choice functions, this idea suggests that every collection of nonempty sets can be associated with a choice function. In the mathematical landscape, AC’s significance is accentuated by its extensive application in a myriad of mathematical deductions, marking it as a cornerstone among mathematical axioms. This study delves into the practical implications and applications of AC, employing rigorous analytical methods to investigate its vast and multifaceted influence on the broader mathematical domain. Key findings indicate that AC has facilitated the proof of various theorems, some of which, on the surface, appear unrelated. While its inception sparked considerable debates due to concerns over its intuitive validity, its impact on contemporary mathematics is profound. This research underscores AC’s central role in advancing mathematical thought, highlighting its contributions to both foundational theories and intricate proofs.
Keywords
Axiom of Choice, Set Theory, Application
References
1. Aczel, P., & Gambino, N. (2002). Collection principles in dependent type theory. In P. Callaghan, Z. Luo, J. McKinna & R. Pollack (Eds.), Types for Proofs and Programs (Lecture Notes on Computer Science, Volume 2277) (pp. 1–23). Springer.
2. Howard, P., & Rubin, J. E. (1998). Consequences of the Axiom of Choice (Vol. 59). American Mathematical Society Surveys and Monographs.
3. Cohen, P.J. (1963). The independence of the continuum hypothesis I. Proceedings of the U.S. National Academy of Sciences, 50(1143), 1143–48.
4. Rubin, H., & Rubin, J. E. (1985). Equivalents of the Axiom of Choice II. North-Holland.
5. Bell, J.L., & Fremlin, D. (1972). The maximal ideal theorem for lattices of sets. Bulletin of the London Mathematical Society, 4(1), 1–2.
6. Post, E.L. (1953). A necessary condition for definability for transfinite von Neumann-Gödel set theory sets, with an application to the problem of the existence of a definable well-ordering of the continuum. Bulletin of the American Mathematical Society, 59(246).
7. Martos-Garcia, D., Devís-Devís, J., & Sparkes, A. C. (2022). Volunteering for research in prison: issues of access, rapport and ethics and emotions during ethnography. International Journal of Qualitative Methods, 21, 16094069221086096.
8. Diekmann, A. (2022). Rational choice sociology: heuristic potential, applications, and limitations. In Handbook of sociological science (pp. 100-119). Edward Elgar Publishing.
9. Kochenderfer, M. J., Wheeler, T. A., & Wray, K. H. (2022). Algorithms for decision making. MIT Press.
10. Fauguel, S. (2023). Male Clients’ Perspective Of Their Experience Of Counselling In Prisons Stephen Brian Fauguel (Doctoral dissertation).
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International 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 Instruction).
- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-233-6
- ISBN (Online)
- 978-1-83558-234-3
- Published Date
- 20 December 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/25/20240842
- Copyright
- 20 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