Is Wikipedia wrong when stating that $\emptyset$ has exactly one partition, namely $\emptyset$?

Yes, the empty set does have a partition. Let's see what a partition is: given a set $X$, a partition of $X$ is a set $P$ of nonempty subsets of $X$ such that each element of $X$ is contained in exactly one element of $P$.

Consider $P = \varnothing$.

• Is it a collection of nonempty subsets of $\varnothing$? Yes, all the elements of $P$ are nonempty subsets of $\varnothing$, because $P$ has no elements so the assertion is vacuously true.

• Is every element of $\varnothing$ contained in exactly one element of $P$? Again yes, this is vacuously true.

Therefore $P = \varnothing$ is indeed a partition of $\varnothing$. However $P = \{ \varnothing \}$, the set with one member, is not a partition of $\varnothing$, because it fails the first requirement.

It's not $\{\emptyset\}$, but $\emptyset$. That is, $P$ (the partition itself) has no elements. This means:

• Every element of $P$ is nonempty (since it doesn't have any).

• Meanwhile, the union of the elements of $P$ is $\emptyset$: every element of $\emptyset$ is in some element of $P$ (since $\emptyset$ doesn't have any elements), and every element of an element of $P$ is an element of $\emptyset$ (since $P$ doesn't have any elements).

So $P$ is indeed a partition of $\emptyset$, albeit in a silly way.