# Empty set

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 $$B$$ (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: 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. noteStrictly, 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.

Parents:

• Empty set

The empty set does what it says on the tin: it is the set which is empty.

• Set

An un­ordered col­lec­tion of dis­tinct ob­jects.