A key notes on Set Theory Origin

Set theory is a branch of mathematics that deals with well-defined groupings of objects known as members or elements of a set. Only sets whose members are also sets are considered in pure set theory. Formally, the theory of hereditarily-finite sets, i.e., finite sets whose elements are also finite sets whose elements are also finite sets, and so on, is identical to arithmetic. Set theory can be defined as the mathematical theory of the observed as opposed to potential—infinite since its essence is the study of infinite sets.

A history of set theory

Georg Cantor’s work is the beginning of set theory as a separate mathematical science. Set theory was conceived in late 1873, when he discovered that the linear continuum, i.e., the real line, is not countable, i.e., its points cannot be counted using natural numbers. Despite the fact that both the set of natural numbers and the set of real numbers are infinite, there are more real numbers than natural numbers, which led to the study of infinity’s various sizes. For a study of the origins of set-theoretic ideas and their application by various mathematicians and philosophers prior to and around Cantor’s work, see the section on the early development of set theory.

1874 and 1897, the German mathematician and logician Georg Cantor created a theory

Set theory is an area of mathematics that studies the features of well-defined groupings of objects, such as numbers or functions, that may or may not be mathematical in nature. The idea is more useful as a foundation for accurate and adaptable terminology for defining complex and sophisticated mathematical topics than as a direct application to everyday experience.

Georg Cantor, a German mathematician and logician, developed an abstract set theory and turned it into a mathematical discipline between the years of 1874 and 1897. This theory arose from his research into a few specific problems involving infinite sets of real numbers of various types. Cantor defined a set as “a group of distinct, discernible objects of sense or thinking understood as a whole.” Members of the set are referred to as elements.

The theory was groundbreaking in that it treated infinite sets as mathematical objects on par with ones that can be built in a finite number of steps. Since antiquity, most mathematicians have avoided including the actual infinite in their arguments (i.e., of sets containing an infinite number of objects conceived as existing simultaneously, at least in thought). Cantor’s work was the subject of much criticism because it dealt with fictions—indeed, it encroached on the domain of philosophers and violated religious principles—as long as this attitude continued until nearly the end of the nineteenth century. However, when further applications to analysis were discovered, attitudes began to shift, and by the 1890s, Cantor’s concepts and findings were becoming widely accepted. Set theory was established as a separate subject of mathematics by 1900.

However, other inconsistencies in what is known as naive set theory were uncovered around the same time. An axiomatic framework for the theory of sets, similar to that developed for elementary geometry, was developed to eliminate such issues. The Nicolas Bourbaki Éléments de mathématique (begun 1939; “Elements of Mathematics”) well expresses the degree of accomplishment that has been achieved in this process, as well as the current status of set theory: “It is now recognised that it is theoretically viable to deduce almost all known mathematics from a single source, The Theory of Sets.”

Conclusion 

Georg Cantor, full name Georg Ferdinand Ludwig Philipp Cantor, was a German mathematician who formed set theory and invented the mathematically meaningful concept of transfinite numbers, which are infinitely large but distinct from one another. He was born on March 3, 1845, in St. Petersburg, Russia, and died on January 6, 1918, in Halle, Germany.

Cantor initially addressed the theory of numbers in a series of ten papers published between 1869 and 1873; this essay represented his own interest in the subject, as well as his studies of Gauss and Kronecker. Cantor next went to the theory of trigonometric series, in which he expanded the concept of real numbers, following the advice of Heinrich Eduard Heine, a colleague at Halle who noticed his ability. Cantor demonstrated in 1870 that a complex variable function can be expressed in only one way by a trigonometric series, based on the work of German mathematician Bernhard Riemann on trigonometric series and the function of a complex variable in 1854. In 1872, he defined irrational numbers in terms of convergent sequences of rational numbers (quotients of integers) and then began his major lifework, the theory of sets and the concept of transfinite numbers, as a result of his consideration of the collection of numbers (points) that would not conflict with such a representation.