# Set theory

Set theory is a branch of mathematics dealing with collections of objects, called sets. It revolutionized mathematics and made possible enormous new insights.

• The language of set theory is based on a single fundamental relation, called membership. We say that x is a member of A (in symbols $x \in A$), or that the set A contains x as an element.[1] The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements,[2] or, equivalently, if each is a subset of the other.

## History of set theory

It was developed in the late 1800s, primarily by the German mathematician Georg Cantor. This initial attempt became known as "naive set theory" because mathematicians found flaws in it. It was replaced by "axiomatic set theory" in the early 1900s. The most commonly used such axiomatization is Zermelo-Fraenkel set theory.

An initial insight of set theory, against intense opposition by established mathematicians, was that some infinities are larger than others. Previously it was thought that infinity had only one size.

One paradox in naive set theory was announced by Bertrand Russell in 1901, and is known as Russell's Paradox.

Like all sufficiently strong mathematical theories, set theory is incomplete, as shown by Kurt Godel. However, set theory is the received axiomatization of mathematics today, with subjects like analysis, algebra, topology, and geometry using set theory and its language for their own foundation.

## The Empty Set

The empty set is the set with no members. Because sets are uniquely defined by membership, the empty set is unique. The empty set is usually denoted by {} or ∅.