It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will Brian, we are discussing some properties of Cohen and random generic reals. One of the properties we are discussing is about definability of $\mathbb{Z}$ after adding a Cohen or a random real.
Let $V[G]$ be a generic extension of $V$ by adding a Cohen (resp. random) real $r$ over $V$. Then we can prove the following:
(1) If we consider $r$ as a constant, then $\mathbb{Z}$ is not definable in $(\mathbb{R}^{V[G]}, +, \cdot, <, 0, 1, r)$, and in fact this structure remains o-minimal (this is a known fact).
(2) If we consider $r \subset \mathbb{N}$ as a predicate, then $\mathbb{Z}$ is definable in $(\mathbb{R}^{V[G]}, +, \cdot, <, 0, 1, r)$ (This is easily seen as $\mathbb{Z}=\{x \in \mathbb{R}^{V[G]}: \exists a, b \in r, x=a - b \}$).
We also have proved the following.
(3) There exists a generic extension $V[G]$ of $V$ by adding a single real $r$, such that considering $r$ as a predicate, $\mathbb{Z}$ is not definable in $(\mathbb{R}^{V[G]}, +, \cdot, <, 0, 1, r)$.
Now the following question arises.
Question. Let $V[G]$ be a generic extension of $V$ by adding a Cohen (resp. random) real over $V$. Let $C=\{x \in \mathbb{R}^{V[G]}: x$ is Cohen (resp. random) over $V \}$. Is $\mathbb{Z}$ definable in $(\mathbb{R}^{V[G]}, +, \cdot, <, 0, 1, C)$.
Remark. The structure $(\mathbb{R}^{V[G]}, +, \cdot, <, 0, 1, C)$ is not o-minimal.
