  Endowed with the binary operation of set addition, the family of all finite
subsets of $\mathbb N$ containing $0$ is itself a monoid, herein denoted by
$\mathcal P_{{\rm fin},0}(\mathbb N)$, whose identity element is the singleton
$\{0\}$. We establish that the only non-trivial automorphism of $\mathcal
P_{{\rm fin},0}(\mathbb N)$ is the involution $X \mapsto \max X - X$. The proof
uses ideas from additive combinatorics and an unconventional induction on what
we call the boxing dimension of a finite set $X \subseteq \mathbb N$, that is,
the smallest number of intervals whose union is $X$.
