POSSÍVEL MÉTODO PARA ENUMERAÇÃO DE ELEMENTOS BINÁRIOS

Um dos métodos utilizados para identificar quais sequências correspondem em bijeção com o conjunto dos Naturais é a Diagonalização, originalmente desenvolvido pelo matemático Georg Cantor. A Diagonalização comumente é considerada como prova, demonstra que o conjunto dos Binários não é enumerável e que a cardinalidade dos Binários e dos Naturais são diferentes. Entretanto neste trabalho é apontado os indícios de um meio de se estabelecer relação de um para um entre elementos Binários e elementos dos Naturais, para tal feito é utilizado princípio da boa ordem, análise combinatória e teoria de conjuntos. Foram estabelecidas duas demonstrações, na primeira foi considerada relevante a quantidade de casas que definem um elemento binário bem como o valor associado, na segunda demonstração foi considerado relevante apenas o valor associado. Nas duas demonstrações atingiu-se a enumeração dos binários. Tais resultados poderiam representar um novo método para enumerar sequências binárias e possivelmente as demais sequências infinitas. Os conjuntos dos Racionais e Naturais também foram estudadas por meio deste método.

The way to list all infinite real numbers and to construct a bijection between natural numbers and real numbers

Georg Cantor defined countable and uncountable sets for infinite sets. The set of natural numbers is defined as a countable set, and the set of real numbers is proved to be uncountable by Cantor’s diagonal argument. Most scholars accept that it is impossible to construct a bijection between the set of natural numbers and the set of real numbers. However, the way to construct a bijection between the set of natural numbers and the set of real numbers is proposed in this paper. The set of real numbers can be proved to be countable by Cantor’s definition. Cantor’s diagonal argument is challenged because it also can prove the set of natural numbers to be uncountable. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

DEL INFINITO POTENCIAL AL ACTUAL: UN RECORRIDO HISTÓRICO A TRAVÉS DE LA METÁFORA CONCEPTUAL

En este artículo abordamos el análisis histórico y epistemológico del infinito como concepto matemático, mirado bajo el lente de la metáfora conceptual, tratando de precisar los obstáculos que impidieron, por largos períodos de nuestra historia, la aceptación del infinito actual, permitiéndose solamente la existencia del infinito potencial. Argumentamos que este análisis es un tema de especial relevancia a considerar en las agendas de investigación dentro de la didáctica de las matemáticas. En particular, mostraremos cómo el desarrollo de ciertas metáforas conceptuales condujo a un proceso de axiomatización del infinito actual, que concluyó con los trabajos de Georg Cantor. La metodología implementada se apoya en una investigación bibliográfica de carácter cualitativo y argumentativo fundamentada en una meta-etnografía. A partir de esta investigación, se obtiene información sobre las estructuras matemáticas que transitan entre los diferentes dominios de partida y de llegada de las metáforas conceptuales, a través de las cuales se desarrolló el infinito matemático durante cuatro estadios principales de la historia, mostrando además la transición del infinito potencial al infinito actual. En particular, se identifican al menos cinco metáforas diferentes que el profesor debe considerar y el estudiante debe desarrollar para lograr una comprensión adecuada del infinito matemático.

A new perspective of countable and uncountable infinite sets on Georg Cantor’s definition in set theory

Georg Cantor defined countable and uncountable sets for infinite sets. Natural number set is defined as a countable set, and real number set is proven as an uncountable set by Cantor’s diagonal method. However, in this paper, natural number set will be proven as an uncountable set using Cantor’s diagonal method, and real number set will be proven as a countable set by Cantor’s definition. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

A new perspective of countable and uncountable infinite set on Georg Cantor’s definition in set theory

Georg Cantor defined countable and uncountable sets for infinite sets. Natural number set is defined as a countable set, and real number set is proven as an uncountable set by Cantor’s diagonal method. However, in this paper, natural number set will be proven as an uncountable set using Cantor’s diagonal method, and real number set will be proven as a countable set by Cantor’s definition. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou

The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infinite” or the set of all that is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the construction of Badiou’s system of mathematical ontology and portrays the relationship between philosophy and mathematics on the basis of an inductive not deductive reasoning.

FROM GEORG CANTOR TO ERNST MACH

In the review, in the context of the ideas of E. Mach, one of the key problems of modern natural science is touched upon: the correlation of “reductionism” and “holism”. If the idea of reductionism was unprecedentedly developed in the framework of the theory of sets and the Bourbaki program, then the Mach principle, as well as other ideas of the philosopher, are only on the rise. Nevertheless, there is a clear realization that the future lies behind these ideas and the book under review in this context seems extremely important.

Hayek's Spiritual Science

This article argues that Hayek's thought had a consistent epistemological core that he developed with the aim of undermining prevailing positivism and replacing it with a metaphysical and spiritualistic philosophy of science. This becomes clear when an intellectual-historical method is used to elucidate Hayek's psychological and methodological works. We see that the approaches and arguments he found most convincing were those of nineteenth-century neo-Kantianisms, Gestalt psychology, vitalism, phenomenology, and theological mathematician Georg Cantor. Hayek thought his spiritual science superior because it explained “the place where the human individual stands in the order of things,” thereby clarifying science's epistemic standpoint, but also its meaning. The article will be of interest to scholars of neoliberalism and contemporary politics because its reading of Hayek suggests that the allegiance between, and apparent attractiveness of, Hayekian and religious conservative thought may have something to do with their common claims to marry order, freedom, and purpose.