A countable set of transcendentals

Question

Let $S$ be a countable subset of $\mathbb{R}$.

Prove that there is a real number $c$, such that $s+c$ is transcendental for all $s\in S$.

Any hint?

Edit: I was trying in vain to solve this by seeking algebraic properties of transcendental numbers.

Answer 1

Hint: Fix a single algebraic number $\alpha.$ How many $c$ are there so that $\alpha = s+c$ for some $s \in S$?