# Portal:Set theory

## Set theory

A Venn diagram illustrating the intersection of two sets.

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

## Selected article

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.

## Selected set

View of the Mandelbrot 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 xn+1=xn2 + 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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 Logic Mathematics Numbertheory Physics Science Set theory Statistics

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