Definition:Set

Definition
A set is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other.

We can think of each set as a single entity in itself, and we can denote it (and usually do) by means of a single symbol.

Cantor defined a set as being:
 * a Many that allows itself to be thought of as a One.

That is, anything you care to think of can be a set. This concept is known as the comprehension principle.

However, there are problems with the comprehension principle. If we allow it to be used without any restrictions at all, paradoxes arise, the most famous example probably being Russell's paradox.

Defining a Set
The elements in a set $S$ are the things that define what $S$ is.

If $S$ is a set, and $a$ is one of the objects in it, we say that $a$ is an element (or member) of $S$, or that $a$ belongs to $S$, or $a$ is in $S$, and we write $a \in S$.

If $a$ is not one of the elements of $S$, then we can write $a \notin S$ and say $a$ is not in $S$.

Thus a set $S$ can be considered as dividing the universe into two parts:
 * all the things that belong to $S$
 * all the things that do not belong to $S$.

Explicit Definition
If there are many elements in a set, then it becomes tedious and impractical to list them all in one big long explicit definition. Fortunately, however, there are other techniques for listing sets.

Also known as
In the original translation by of 's original work, this concept was called an aggregate. The term can be seen in subsequent works, but has now mostly been superseded by the term set.

Sometimes the terms class, family or collection are used. In some contexts, the term space is used. However, beware that these terms are usually used for more specific things than just as a synonym for set.

On this website, the terms class, family and space are not used as synonyms for set, being reserved specifically for the concepts to which they apply.

A set whose elements are all (geometric) points is often called a point set.

Historical Note
Although the concept of a set as currently understood originates mainly with, it first appears in 's posthumous (1851) work Paradoxien des Unendlichen (The Paradoxes of the Infinite).