Axiomatic set theory

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.

Related Topics:
Mathematics - Mathematician - Georg Cantor - 19th century - Foundational theory

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Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rigor in proofs. At the same time the basic concepts of set theory are used throughout mathematics, the subject is pursued in its own right as a specialty by a comparatively small group of mathematicians and logicians.

Related Topics:
Mathematical rigor - Logic

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It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics.

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The basic concepts of set theory are set and membership. A set is thought of as any collection of objects, called the members (or elements) of the set. In mathematics, the members of sets are any mathematical objects, and in particular can themselves be sets. Thus one speaks of the set N of natural numbers { 0, 1, 2, 3, 4, ... }, the set of real numbers, and the set of functions from the natural numbers to the natural numbers; but also, for example, of the set { 0, 2, N } which has as members the numbers 0 and 2 and the set N.

Related Topics:
Set - Natural number - Real number - Function

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Initially, what is now known as "naive" or "intuitive" set theory was developed. (See naive set theory). As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox. To address these problems, set theory had to be re-constructed, this time using an axiomatic approach.

Related Topics:
Naive set theory - Paradox - Russell's paradox - Axiom

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