Definition:Universe (Set Theory)

Definition
Sets are considered to be subsets of some large universal set, also called the universe.

Exactly what this universe is will vary depending on the subject and context.

When discussing particular sets, it should be made clear just what that universe is.

However, note that from There Exists No Universal Set, this universe cannot be everything that there is.

The traditional symbol used to signify the universe is $\mathfrak A$.

However, this is old-fashioned and inconvenient, so some newer texts have taken to using $\mathbb U$ or just $U$ instead.

With this notation, this definition can be put into symbols as:
 * $\forall S: S \subseteq \mathbb U$

The use of $\mathbb U$ or a variant is not universal: some sources use $X$.

Also known as
Some sources refer to the universal set as the universe of discourse, which name is also used for a similar concept in the category of predicate logic.

Some sources (particularly in the context of probability theory) refer to it as the space.

Also see

 * Definition:Universe of Discourse
 * Definition:Universal Class
 * Definition:Population


 * There Exists No Universal Set