# Empty set

The empty set is the set hav­ing no mem­bers. It is usu­ally de­noted as $$\emptyset$$. What­ever ob­ject is con­sid­ered, it can’t be a mem­ber of $$\emptyset$$. It might be use­ful in the be­gin­ning to think about the empty set as an empty box. It has noth­ing in­side it, but it still does ex­ist.

For­mally, the ex­is­tence of the empty set is as­serted by the Empty Set Ax­iom:

$$\exists B \forall x : x∉B$$

The empty set ax­iom it­self does not pos­tu­late the unique­ness of $$\emptyset$$. How­ever, this fact is easy to prove us­ing the ax­iom of ex­ten­sion­al­ity. Con­sider sets $$A$$ and $$B$$ such that both $$\forall x : x∉A$$ and $$\forall x: x∉B$$. noteThat is, sup­pose we had two empty sets. Re­mem­ber that the ex­ten­sion­al­ity ax­iom tells us that if we can show $$\forall x : (x ∈ A \Leftrightarrow x ∈ B)$$, then we may de­duce that $$A=B$$. In this case, for ev­ery $$x$$, both parts of the state­ment $$(x ∈ A \Leftrightarrow x ∈ B)$$ are false: we have $$x \not \in A$$ and $$x \not \in B$$. There­fore the iff re­la­tion is true.

The ex­is­tence of the empty set can be de­rived from the ex­is­tence of any other set us­ing the ax­iom schema of bounded com­pre­hen­sion, which states that for any for­mula $$\phi$$ in the lan­guage of set the­ory, $$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \phi(x))$$. In par­tic­u­lar, tak­ing $$\phi$$ to be $$\bot$$, the always-false for­mula, we have that $$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \bot)$$. Since $$x \in b \Leftrightarrow (x \in a \wedge \bot)$$ is log­i­cally equiv­a­lent to $$x \in b \Leftrightarrow \bot$$ and hence to $$x \notin b$$, the quan­tified state­ment is log­i­cally equiv­a­lent to $$\forall a \exists b \forall x : x \notin b$$, and as soon as we have the ex­is­tence of at least one set to use as $$a$$, we ob­tain the Empty Set Ax­iom above.

It is worth not­ing that the empty set is it­self a sin­gle ob­ject. One can con­struct a set con­tain­ing the empty set: $$\{\emptyset\}$$. $$\{\emptyset\} \not= \emptyset$$, be­cause $$\emptyset ∈ \{\emptyset\}$$ but $$\emptyset ∉ \emptyset$$; so the two sets have differ­ent el­e­ments and there­fore can­not be equal by ex­ten­sion­al­ity. noteIn terms of the box metaphor above, $$\{\emptyset\}$$ is a box, con­tain­ing an empty box, whilst $$\emptyset$$ is just an empty box

Another way to think about this is us­ing car­di­nal­ity. In­deed, $$|\{\emptyset\}| = 1$$ (as this set con­tains a sin­gle el­e­ment - $$\emptyset$$) and $$|\emptyset| = 0$$ (as it con­tains no el­e­ments at all). Con­se­quently, the two sets have differ­ent amounts of mem­bers and can not be equal.

the empty set is of­ten used to rep­re­sent the or­di­nal 0

Punc­tu­a­tion can be weird in this edit, as the au­thor is not a na­tive English speaker. Might need to be improved

Children:

Parents:

• Set

An un­ordered col­lec­tion of dis­tinct ob­jects.

• @5hc Thanks for the edit! I made a cou­ple of lin­guis­tic changes, and made the “unique­ness of $$\emptyset$$” a bit less com­pact.