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Naive set theory was created at the end of the 19th century by Georg Cantor in order to allow mathematicians to work with infinite sets consistently.
As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox. In response, axiomatic set theory was developed to determine precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually mean axiomatic set theory, but when they talk about set theory as a mere tool to be applied to other mathematical fields, they usually mean naive set theory.
Axiomatic set theory can be quite abstruse and yet has little effect on ordinary mathematics. Thus, it is useful to study sets in the original naive sense in order to develop facility for working with them. Furthermore, a firm grasp of naive set theory is important as a first stage in understanding the motivation for the axiomatic theory.
This article develops the naive theory. Sets are defined informally and a few of their properties are investigated. Links in this article to specific axioms of set theory point out some of the relationships between the informal discussion here and the formal axiomatization of set theory, but no attempt is made to justify every statement on such a basis.
In naive set theory, a set is described as a collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. As can be seen from this example, sets are allowed to have an infinite number of elements.
If x is a member of A, then it is also said that x belongs to A, or that x is in A. In this case, we write x ∈ A. (The symbol "∈" is a derivation from the Greek letter epsilon, "ε", introduced by Peano in 1888.)
Two sets A and B are defined to be equal when they have precisely the same elements, that is, if every element of A is an element of B and every element of B is an element of A. (See axiom of extensionalitySet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality or axiom of extension is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Fraen.) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numberIn mathematics, a prime number or prime for short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number . Hence a prime number has exactly two divisors. The number 1 is neither prime nor coms less than 6. If A and B are equal, then this is denoted symbolically as A = B (as usual).
We also allow for an empty setAbstract algebra Algebra Set theory In mathematics, the empty set is the set with no elements. Notation The standard notation for denoting the empty set, invented by Nicholas Bourbaki, is the symbol , also written as or ∅, and sometimes approximated, a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty setSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Frankel axioms, the axiom reads:.)
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