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Set theory

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Set theory is the branch of mathematics that studies sets, which are collections of distinct objects. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the most well known.

Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects

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A set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental concepts in mathematics and their formalization at the end of the 19th century was a major event in the history of mathematics and lead to the unification of a number of different areas. The idea of function comes along naturally, as "morphisms" between sets.

The study of the structure of sets, set theory, can be viewed as a foundational ground for most of mathematical theories. Sets are usually defined axiomatically using an axiomatic set theory, this way to study sets was introduced by Georg Cantor between 1874 and 1884 and deeply inspired later works in logic. Sets are representable in the form of Venn diagrams, for instance it can represent the idea of union, intersection and other operations on sets.

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In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883[1] (but discovered in 1875 by Henry John Stephen Smith [2]), is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

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Georg Cantor (March 3, 1845 – January 6, 1918) was a German mathematician. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

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The intersection of two sets is the set that contains all elements of one of these sets that also belong to the other one, but no other elements. It is possible to define the intersection of several sets, and even of an infinite family of sets.  

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  1. ^ Georg Cantor, On the Power of Perfect Sets of Points (De la puissance des ensembles parfait de points), Acta Mathematica 4 (1884) 381--392. English translation reprinted in Classics on Fractals, ed. Gerald A. Edgar, Addison-Wesley (1993) ISBN 0-201-58701-7
  2. ^ Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos