# 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

## Selected article

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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## Selected set

In mathematics, the **Mandelbrot set** is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex *c*-values for which the orbit of 0 under iteration of the complex quadratic polynomial *x*_{n+1}=*x*_{n}^{2} + *c* remains bounded. When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition.

## Selected biography

**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.

## Did you know?

- ... that there exists a
**composition of binary relations**consistent with the composition of functions ? - ... that there is an
**ordinal arithmetic**extending the arithmetic of natural numbers to the ordinal numbers ? - ... that sets which are both infinite and countable have a cardinality of
**aleph null**?

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*Topics in Set Theory*

Key concepts | Types of set theory | Mathematical logic | Set-theoretic constructions |
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