Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / B / 5 / iv

Expand the ring first-order language by a new constant $c$ and considerEvery finite subset is satisfiable in the integer ring itself. Only finitely many divisibility conditions occur; interpret $c$ as a nonzero common multiple of their positive integer indices, using $1$ if none occur. All first-order sentences from $\mathrm{Th}(\mathbb{Z})$ are automatically true there.

Compactness theorem therefore gives a first-order model. Its reduct $R$ to the original ring first-order language satisfies the complete theory of $\mathbb{Z}$, so $R\equiv \mathbb{Z}$. Its element $c^R$ is nonzero and divisible by every positive standard integer. ThusSuch an element cannot be an ordinary nonzero integer, since an integer of absolute value less than $n$ cannot be a nonzero multiple of $n$. The statement concerns each standard integer index separately; it does not assert divisibility by every internally nonzero element of $R$.