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.


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.


  • Empty set

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