The origin of “Zorn's Lemma”

Campbell, Paul J.

doi:10.1016/0315-0860(78)90136-2

Cited by 44 publications

(5 citation statements)

References 13 publications

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“…Actually, we are unable to construct any countable set of vectors in  ∞ that may span it. Although, Zorn's Lemma can be applied to show that every V has bases, see [2][3][4]. However, it does not explain the procedure of how to actually construct these bases if V is not finitely generated.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal Polynomials Based Operational Matrices with Applications to Bagley-Torvik Fractional Derivative Differential Equations

Talib¹,

Özger²

2023

Recent Research in Polynomials [Working Title]

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Orthogonal polynomials are the natural way to express the elements of the inner product spaces as an infinite sum of orthonormal basis sets. The construction and development of the many important numerical algorithms are based on the operational matrices of orthogonal polynomials including spectral tau, spectral collocation, and operational matrices approach are few of them. The widely used orthogonal polynomials are Legendre, Jacobi, and Chebyshev. However, only a few papers are available where the Hermite polynomials (HPs) were exploited to solve numerically the differential equations. The notable characteristic of the HPs is its ability to approximate the square-integrable functions on the entire real line. The prime objective of this chapter is to introduce the two new generalized operational matrices of HPs which are developed in the sense of the Riemann-Liouville fractional-order integral operator and Hilfer fractional-order derivative operator. The newly derived operational matrices are further used to construct a numerical algorithm for solving the Bagley--Trovik types fractional derivative differential equations (FDDE). Moreover, the results obtained by using the proposed algorithm are compared with the results obtained otherwise to demonstrate the efficiency and accuracy of the proposed numerical algorithm. Some examples are solved for application purposes.

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Section: Introductionmentioning

confidence: 99%

Orthogonal Polynomials Based Operational Matrices with Applications to Bagley-Torvik Fractional Derivative Differential Equations

Talib¹,

Özger²

2023

Recent Research in Polynomials [Working Title]

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“…are barred, from the algebraic point of view, by the well-ordering theorem and its theory. (Zorn 1935, 667) As a way out he coined what is now known as "Zorn's Lemma" (ZL) (Zorn 1935;Campbell 1978), to have a variant of WO more palatable to algebraists. Since then, ZL has gradually replaced WO in proof practice, especially in algebra.…”

Section: Introductionmentioning

confidence: 99%

Syntax for Semantics: Krull’s Maximal Ideal Theorem

Schuster

Wessel

2021

Paul Lorenzen -- Mathematician and Logician

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Krull’s Maximal Ideal Theorem (MIT) is one of the most prominent incarnations of the Axiom of Choice (AC) in ring theory. For many a consequence of AC, constructive counterparts are well within reach, provided attention is turned to the syntactical underpinning of the problem at hand. This is one of the viewpoints of the revised Hilbert Programme in commutative algebra, which will here be carried out for MIT and several related classical principles.

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“…Mais tarde, uma carta abalou a sociedade Matemática das Américas, pois a mesma trazia a notícia de que o Lema da Zorn não possuía o nome de seu verdadeiro descobridor. Nesta busca pelas origens do lema de Zorn, foram encontrados documentos que comprovam, além de contribuição de diversos autores, o seu uso anterior as pesquisas de Zorn (Campbell, 1978).…”

Section: Introductionunclassified

“…Segundo (Campbell, 1978), o lema conhecido atualmente como" Lema de Zorn" já havia sido citado por outros autores, como: (Rosser, 1953, Cuesta, 1955, Fraenkel & Bar-Hillel, 1958, Rubin & Rubin, 1963, Beth, 1964, Semadeni, 1968, Jech 1973& Suppers, 1960. Muitos desses autores acabaram por estudar o tal Lema de Zorn, a história escondida por trás do nome que chama atenção, pois Max já conhecia relatos de outros estudos sobre o Lema.…”

Section: Introductionunclassified

O lema Kuratowski-Zorn na matemática do Ensino Médio

Rosário

Balieiro

Dias

et al. 2021

RSD

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O Lema Kuratowski-Zorn denominado frequentemente apenas por Lema de Zorn é objeto de controvérsias e debates desde o seu surgimento, apesar disto, sua relevância pode ser percebida pela grande quantidade de aplicações e consequências apresentadas sobre a forma de enunciados equivalentes que são presentes na área de exatas, e também em aplicações matemáticas que se baseiam na teoria dos conjuntos como, axioma da Escolha, Teorema da Boa ordem, Teorema de Tychonoff, Teorema Hahn Banach entre outros, sendo os três primeiros equivalentes ao Lema de Zorn. Por este motivo, teremos como objetivo, partindo de uma abordagem histórica sobre o mesmo, apresentar umas das aplicações que são responsáveis por fortalecer sua notoriedade e popularidade entre estudiosos matemáticos e cientistas das mais variadas áreas que se utilizam da Álgebra Linear para representar e resolver seus modelos matemáticas utilizando este Lema e suas equivalências, onde veremos como podemos encontrá-lo na matemática do ensino médio regular das escolas quando apresentada aos alunos.

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The origin of “Zorn's Lemma”

Cited by 44 publications

References 13 publications

Orthogonal Polynomials Based Operational Matrices with Applications to Bagley-Torvik Fractional Derivative Differential Equations

Orthogonal Polynomials Based Operational Matrices with Applications to Bagley-Torvik Fractional Derivative Differential Equations

Syntax for Semantics: Krull’s Maximal Ideal Theorem

O lema Kuratowski-Zorn na matemática do Ensino Médio

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