NOTE ON POINT SET THEORY

The term ‘mathematical analysis’ refers to the major branch of mathematics which is concerned with the theory of functions and includes the differential and integral calculus. Analysis and the calculus began as the study of curves, calculus being concerned with tangents to and areas under curves. The focus was shifted to functions following the insight, due to Leibniz and Isaac Newton in the second half of the seventeenth century, that a curve is the graph of a function. Algebraic foundations were proposed by Lagrange in the late eighteenth century; assuming that any function always took an expansion in a power series, he defined the derivatives from the coefficients of the terms. In the 1820s his assumption was refuted by Cauchy, who had already launched a fourth approach, like Newton’s based on limits, but formulated much more carefully. It was refined further by Weierstrass, by means which helped to create set theory. Analysis also encompasses the theory of limits and of the convergence and divergence of infinite series; modern versions also use point set topology. It has taken various forms over the centuries, of which the older ones are still represented in some notations and terms. Philosophical issues include the status of infinitesimals, the place of logic in the articulation of proofs, types of definition, and the (non-) relationship to analytic proof methods.

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Questions of generality as probes into nineteenth-century mathematical analysis

10.1093/oxfordhb/9780198777267.013.14 ◽

2017 ◽

Author(s):

Renaud Chorlay

Keyword(s):

Nineteenth Century ◽

Set Theory ◽

Mathematical Analysis ◽

Analytic Functions ◽

Function Theory ◽

The Third ◽

Point Set

This article examines ways of expressing generality and epistemic configurations in which generality issues became intertwined with epistemological topics, such as rigor, or mathematical topics, such as point-set theory. In this regard, three very specific configurations are discussed: the first evolving from Niels Henrik Abel to Karl Weierstrass, the second in Joseph-Louis Lagrange’s treatises on analytic functions, and the third in Emile Borel. Using questions of generality, the article first compares two major treatises on function theory, one by Lagrange and one by Augustin Louis Cauchy. It then explores how some mathematicians adopted the sophisticated point-set theoretic tools provided for by the advocates of rigor to show that, in some way, Lagrange and Cauchy had been right all along. It also introduces the concept of embedded generality for capturing an approach to generality issues that is specific to mathematics.

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CHAPTER I . POINT SET THEORY

Functional Operators (AM-21), Volume 1 ◽

10.1515/9781400881895-002 ◽

1950 ◽

pp. 1-11

Keyword(s):

Set Theory ◽

Point Set

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Non-Baire Unions in Category Bases

Georgian Mathematical Journal ◽

10.1515/gmj.1996.543 ◽

1996 ◽

Vol 3 (6) ◽

pp. 543-546

Author(s):

J. Hejduk

Keyword(s):

Set Theory ◽

Sufficient Condition ◽

Point Set

Abstract We consider a partition of a space X consisting of a meager subset of X and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset of X. The condition is formulated in terms of the theory of J. Morgan [Point Set Theory, Marcel Dekker, 1990].

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On Soft Real Point Matrices and Their Operations

Moroccan Journal of Pure and Applied Analysis ◽

10.1515/mjpaa-2018-0002 ◽

2018 ◽

Vol 4 (1) ◽

pp. 9-16 ◽

Cited By ~ 2

Author(s):

S. Hussain ◽

H. F. Akiz ◽

A. I. Alajlan

Keyword(s):

Set Theory ◽

Real Point ◽

Theoretical Studies ◽

Point Set

AbstractIn this paper, we define and investigate soft real point matrices and their operations which are more functional to make theoretical studies in the soft real point set theory. We then define products of soft real point matrices and their properties. Examples are also provided to validate the existence of defined notions.

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