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 {curly brackets}, and separates them with commas. For example. the set $\{1,2,3\}$ holds 1, 2, and 3. Sets are also often referred to using capital roman letters such as $A$, $B$, $C$.

A Venn diagram illustrating the intersection of two sets.

There are three methods one can use to describe a set: Description method, Roster method (tabular form) and Rule method (set builder form).

History

Set theory was created around 1874 by Georg Cantor. It had to be made better because collections of objects can cause problems if we 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 was 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 (proper) 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 $o\in A$. 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 $A\subseteq B$. 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 not to have this, the term proper subset is meant not to have this possibility.

The self-considering objects in set theory was considered too, with some example numbers being 1={1}, 2={1, 2}, 3={1, 2, 3} and so on.

Related pages

• Cantor set
• Cantor's theorem
• Cardinality
• Continuum (mathematics)
• Mathematical logic
• Naive set theory
• Zermelo–Fraenkel set theory

Further reading

• Chechulin V. L., Theory of sets with self-considering (foundations and some applications), Publishing by Perm State University (Russia), Perm, 2010, 100 p. ISBN 978-5-7944-1468-4 (In this non-predicative 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 )