  This paper gives a complete proof of a theorem of de Bruijn that classifies
additive systems for the nonnegative integers, that is, families $\mca =
(A_i)_{i\in I}$ of sets of nonnegative integers, each set containing 0, such
that every nonnegative integer can be written uniquely in the form $\sum_{i\in
I} a_i$ with $a_i \in A_i$ for all $i$ and $a_i \neq 0$ for only finitely many
$i$. All indecomposable additive systems are determined.
