$\mathbb{C}_{\exp}$ is not model complete.

Question

I am trying to understand the following proof (Proposition 1.1 here):

Proposition: $\mathbb{C}_{\exp}$ is not model complete.
Proof: If $\mathbb{C}_{\exp}$ is model complete, then every definable set is a projection of a closed set. Since $\mathbb{C}$ is locally compact, every definable set is $F_σ$. The same is true for the complement, so every definable set is also $G_δ$. But, since $\mathbb{Z}$ is definable, $\mathbb{Q}$ is definable and a standard corollary of the Baire Category Theorem tells us that $\mathbb{Q}$ is not $G_δ$.

How can we say that every definable set is $F_σ$?

Here, $\mathbb{C}_{\exp}$ is the field of complex numbers defined in the language of rings with an additional function for exponentiation.

Answer 1

Start with: every definable set is a projection of a closed set. [This is given, and not proved here.]

In $\mathbb C$, every closed set is a union of countably many compact sets. The projection of a compact set is compact. A compact set is closed.

Thus: a projection of a closed set is a countable union of closed sets.

Conclusion: every definable set is an $F_\sigma$.