[summary: The empty set, $~$\emptyset$~$, is the set with no elements. For every object $~$x$~$, $~$x$~$ is not in $~$\emptyset$~$.]

The empty set, $~$\emptyset$~$, is the set with no elements. Why, a priori, should this set exist at all? Well, if we think of sets as "containers of elements", the idea of an empty container is intuitive: just imagine a box with nothing in it.

Definitions

Definition in [ZF]

In the set theory [ZF], there are exactly two axioms which assert the existence of a set ex nihilo; all of the rest of set theory builds sets from those given sets, or else postulates the existence of more sets.

There is the [-axiom_of_infinity], which asserts the existence of an infinite set, and there is the [-empty_set_axiom], which asserts that an empty set exists.

In fact, we can deduce the existence of an empty set even without using the empty set axiom, as long as we are allowed to use the [-axiom_of_comprehension] to select a certain specially-chosen subset of an infinite set. More formally: let $~$X$~$ be an infinite set (as guaranteed by the axiom of infinity). Then select the subset of $~$X$~$ consisting of all those members $~$x$~$ of $~$X$~$ which have the property that $~$x$~$ contains $~$X$~$ and also does not contain $~$X$~$.

There are no such members, so we must have just constructed an empty set.

Definition by a universal property

The empty set has a definition in terms of a Universal property: the empty set is the unique set $~$X$~$ such that for every set $~$A$~$, there is exactly one map from $~$X$~$ to $~$A$~$. More succinctly, it is the [-initial_object] in the [-category_of_sets] (or in the Category of finite sets).

Uniqueness of the empty set

The [-axiom_of_extensionality] states that two sets are the same if and only if they have exactly the same elements. If we had two empty sets $~$A$~$ and $~$B$~$, then certainly anything in $~$A$~$ is also in $~$B$~$ ([vacuous_truth vacuously]), and anything in $~$B$~$ is also in $~$A$~$, so they have the same elements. Therefore $~$A = B$~$, and we have shown the uniqueness of the empty set.

A common misconception: $~$\emptyset$~$ vs $~$\{ \emptyset \}$~$

It is very common for people to start out by getting confused between $~$\emptyset$~$ and $~$\{\emptyset\}$~$. The first contains no elements; the second is a set containing exactly one element (namely $~$\emptyset$~$). The sets don't biject, because they are finite and have different numbers of elements.

[vacuous_truth Vacuous truth]: a whistlestop tour

The idea of vacuous truth can be stated as follows:

For any property $~$P$~$, everything in $~$\emptyset$~$ has the property $~$P$~$.

It's a bit unintuitive at first sight: it's true that everything in $~$\emptyset$~$ is the Pope, for instance. Why should this be the case?

In order for there to be a counterexample to the statement that "everything in $~$\emptyset$~$ is the Pope", we would need to find an element of $~$\emptyset$~$ which was not the Pope. %%note: Strictly, we'd only need to show that such an element must exist, without necessarily finding it.%% But there aren't any elements of $~$\emptyset$~$, let alone elements which fail to be the Pope. So there can't be any counterexample to the statement that "everything in $~$\emptyset$~$ is the Pope", so the statement is true.