Kuratowski-Zorn lemma in High Scholl mathematics

Authors

DOI:

https://doi.org/10.33448/rsd-v10i4.12501

Keywords:

Zorn's lemma; Set theory; Linear algebra; Axiom of choice.

Abstract

Zorn's lemma has been the subject of controversy and debate since its emergence, despite its relevance can be perceived by the large number of applications and consequences presented on the form of equivalent statements that are present in the area of ​​exact, and also in mathematical applications that they are based on the set theory as, axiom of Choice, Theorem of Good Order, Tychonoff's Theorem, Hahn Banach Theorem among others, the first three being equivalent to Zorn's Lemma. For this reason, we will aim, starting from a historical approach on the same, to present one of the applications that are responsible for strengthening its notoriety and popularity among mathematical scholars and scientists from the most varied areas that use Linear Algebra to represent and solve their problems. mathematical models using this lemma and its equivalences, where we will see how we can find it in the mathematics of regular high school schools when presented to students.

Author Biographies

Eduardo da Conceição Rosário, Instituto Federal de Educação, Ciência e Tecnologia do Amapá

Lecturer at the Federal Institute of Education, Science and Technology of Amapá - IFAP, Master's student in Computational Mathematics at Universidade Estadual Paulista Júlio Mesquita Filho (UNESP-PP), Specialization in Management and Teaching in Higher Education (FATECH) Degree in Mathematics from UNIFAP (FEDERAL UNIVERSITY OF AMAPÁ (2012-2016)).

Simone de Almeida Delphim Leal, Universidade Federal do Amapá

Graduated in Mathematics, holds a master's and doctorate in Computational Modeling from the National Laboratory for Scientific Computing (LNCC / RJ) and a post-doctorate at the Institute of Pure and Applied Mathematics (IMPA). Currently at the Federal University of Amapá developing research in optimization and stabilized numerical methods applied to the diffusive equation and in Mathematics Education aimed at the necessary training for the Professional Master in Mathematics.

References

Beth, E (1964) The Foundations of Mathematics: A Study in the Philosophy of Science rev. ed. Harper Torchbooks.

Boccato, V. R. C. (2006) Metodologia da pesquisa bibliográfica na área odontológica e o artigo científico como forma de comunicação. Rev. Odontol. Univ. Cidade São Paulo, 18(3).

Celestino, R. (2000). Ensino-Aprendizagem da Álgebra linear: As pesquisas brasileiras na década de 90. (Dissertação de Mestrado, PUC-SP)

Campbell, P. J. (1978). The origin of Zorn’s Lemma.Historia Mathematica. 77 (85), 77-89.

Cuesta, N 1955 Injusta atribucidn a Zorn de1 principio maximal Gac. Mat. (Madrid) (1) 7, 174-176 (MR 17, 931)

Diniz, M (2019). Teoria dos conjuntos: Sim ou Não? http://mathema.com.br/reflexoes/teoria-dos-conjuntos-sim-ou-nao-2/.

Fraenkel, A. A. & Bar-Hillel, Y. (1958) Foundations of Set Theory Amsterdam (North-Holland) (MR 21, #648)

Grace, A. K. S. G. S. (2010). Infinitos, Contínuos e Escolha: Teoria dos Conjuntos. (Dissertação de Conclusão de Curso, Universidade Federal de São Carlos)

Halmos, P. R. H. R. (2001). Teoria Ingênua dos Conjuntos. Editora Ciência Moderna.

Jech, T. J. (1973). The Axiom of Choice Amsterdam (North- Holland)

Lipschutz, S (1972). Teoria dos Conjuntos. McGraw-Hill Ltda.

Pereira, A. C. C.& Saito, F. (2018). Os instrumentos matemáticos na interface entre história e ensino de matemática: compreendendo o cenário nacional nos últimos 10 anos. Boletim Cearense de Educação e História da Matemática, 5(14).

Rubin, H & Rubin, J. R. H. (1963). Equivalents of the Axiom of choice. Noth-Holland Publishing company-Amsterdam.

Semadeni, Z. (1968) Notes to J. Mikusifiski's article "On Zorn's theorem" (Polish) Wiadom. Mat. (2) 10, 145-146 (MR 39, #1273) (Passages quoted were translated by the author, with the assistance of Ann E. Lynch)

Silva, S. G., & Jesus, J. P. C. (2007). Cem anos do axioma de escolha. Revista Matemática Universitária n◦42.

Suppes, P. (1960) Axiomatic Set Theory Princeton (Van Nostrand)

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Published

09/04/2021

How to Cite

ROSÁRIO, E. da C. .; BALIEIRO, C. A. B. .; DIAS, N. L.; LEAL, S. de A. D. . Kuratowski-Zorn lemma in High Scholl mathematics. Research, Society and Development, [S. l.], v. 10, n. 4, p. e24810412501, 2021. DOI: 10.33448/rsd-v10i4.12501. Disponível em: https://rsdjournal.org/index.php/rsd/article/view/12501. Acesso em: 14 nov. 2024.

Issue

Vol. 10 No. 4

Section

Exact and Earth Sciences

License

Copyright (c) 2021 Eduardo da Conceição Rosário; Cassio Alessandro Batista Balieiro; Neylan Leal Dias; Simone de Almeida Delphim Leal

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