  Let $k,\ell\geq 2$ be two multiplicatively independent integers. Cobham's
famous theorem states that a set $X\subseteq \mathbb{N}$ is both
$k$-recognizable and $\ell$-recognizable if and only if it is definable in
Presburger arithmetic. Here we show the following strengthening: let
$X\subseteq \mathbb{N}^m$ be $k$-recognizable, let $Y\subseteq \mathbb{N}^n$ be
$\ell$-recognizable such that both $X$ and $Y$ are not definable in Presburger
arithmetic. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is
undecidable. This is in contrast to a well-known theorem of B\"uchi that the
first-order logical theory of $(\mathbb{N},+,X)$ is decidable.
