Portal:Set theory
Set theory
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 in the definitions of 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.
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In mathematics, a function is a way to assign to each element of a given set exactly one element of another given set. Functions can be abstractly defined in set theory as a functional binary relation between two sets, respectively the domain and the target of the function. There are many ways to give a function, generally using predefined functions, defined for example in an axiomatic setting. Typically, functions are expressed by a formula, by a plot or graph, by an algorithm that computes it or by a description of its properties. Sometimes, a function is described through its relationship to other functions (see, for example, inverse function).
One idea of enormous importance in all of mathematics is composition of functions, intuitively: if z is a function of y and y is a function of x, then z is a function of x. The existence of identity functions and some basic properties of functions shows that the class of sets forms a category with functions as morphisms.
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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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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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- ^ 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
- ^ Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos