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.

翻译：设 $k,\ell\geq 2$ 为两个乘法无关的整数。科汉著名定理指出：一个集合 $X\subseteq \mathbb{N}$ 既是 $k$-可识别的又是 $\ell$-可识别的，当且仅当它在普雷斯伯格算术中可定义。本文证明如下强化结论：设 $X\subseteq \mathbb{N}^m$ 是 $k$-可识别的，$Y\subseteq \mathbb{N}^n$ 是 $\ell$-可识别的，且 $X$ 与 $Y$ 均不能在普雷斯伯格算术中定义。那么 $(\mathbb{N},+,X,Y)$ 的一阶逻辑理论是不可判定的。这一结果与比希关于 $(\mathbb{N},+,X)$ 的一阶逻辑理论可判定的著名定理形成对比。