# Portal:Set theory

## Set theory

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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In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers. His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. It states:

There is no set whose size is strictly between that of the integers and that of the real numbers.

In light of Cantor's theorem that the sizes of these sets cannot be equal, this hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. The name of the hypothesis comes from the term the continuum for the real numbers.

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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 union of two sets is the set that contains everything that belongs to any of the sets, but nothing else. It is possible to define the union 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