Portal:Set theory
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Set theory is the branch of mathematics that studies sets, which are collections of 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 Cantor and 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.
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 funtions, defined for example in an axiomatic setting. Tipically, 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. For a special treatment of functions in a set-theoritical setting, see function (set theory).
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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.
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.
- ... that there exists a composition of binary relations consistent with the composition of functions ?
- ... that there is an ordinal arithmetic extending the arithmetic of integers to the ordinal numbers ?
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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