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.

Uniqueness of Elements
A set is uniquely determined by its elements. This means that the only thing that defines what a set is is what it contains. So, how you choose to list or define the contents makes no difference to what the contents actually are.

Note these points:

For a given set, an object is either in the set or not in the set. So, if an element is in a set, then it is in the set only once, however many times it may appear in the definition of the set. Thus, the set $\left\{{1, 2, 2, 3, 3, 4}\right\}$ is the same set as $\left\{ {1, 2, 3, 4}\right\}$. $2$ and $3$ are in the set, and listing them twice makes no difference to the set contents. Like the membership of a club, if you're in, you're in - however many membership cards you have to prove it.

It makes no difference what order the elements are specified. This means that the sets $S = \left\{{1, 2, 3, 4}\right\}$ and $T = \left\{{3, 4, 2, 1}\right\}$ are the same set.

Two sets which have exactly the same elements are the same, whatever the sets are called. So, to take the club membership analogy, if two clubs had exactly the same members, the clubs would be considered as the same club, although they may be given different names. This follows from the definition of equals given above.

Note that there are mathematical constructs which do take into account both (or either of) the order in which the elements appear, and the number of times they appear, but these are not sets as such.

Also known as
In the original translation by Jourdain of Georg Cantor'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 Georg Cantor, it first appears in Bolzano's posthumous (1851) work Paradoxien des Unendlichen (The Paradoxes of the Infinite).