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Set theory
Set theory is the study of sets in mathematics. Sets are collections of objects. We refer to these objects as "elements" or "members" of the set. To write a set, one wraps the numbers in {curley brackets} and separates them with commas: eg., {1, 2, 3} holds 1, 2, and 3.
History
Set theory was made in 1874 by Georg Cantor. It had to be made better because collections of objects can cause problems if you work with them without explaining them better. Russell's paradox was one of the problems. Think about the set of all sets that are not members of themselves. If it were inside itself, then the rule which defines it would mean that it is not inside itself. But, if it were not inside itself, then the rule which defines it would mean that it is inside itself. This is a serious problem, and it meant that the old set theory was broken. It was improved by people including Zermelo and Bertrand Russell.
Theory
Set theory begins by giving some examples of things that are sets. Then it gives rules in which you can make other sets from the already known sets. Collections of objects that are not sets are called classes. It is possible to do mathematics using only sets, rather than classes, so that the problems that classes cause in mathematics do not occur.
 Example: An object o and a set A. If o is a member (or element) of A, we write o ∈ A. Since sets are objects, the membership relation can relate sets as well.
A binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, marking A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this example, it is clear that a set is a subset of itself. In cases where one wishes to not to have this, the term proper subset is meant not to have this possibility.
The selfconsidering object in the set theory was existing too, an example numbers: 1={1}, 2={1, 2}, 3={1, 2, 3} and so on.
 Chechulin V. L., Theory of sets with selfconsidering(foundations and some applications), Publishing by Perm State University (Russia), Perm, 2010, 100 p. ISBN 9785794414684 (In this nonpredicative theory overcoming the limitations of predicative formal systems, i.e. Godel's theorems, the text is accessible from the page http://elibrary.ru/item.asp?id=15267103 )

