Is the intersection of infinitely many recursive sets recursive?
For each $i\in \mathbb{N}$, take $S_i = \mathbb{N}\setminus \{i\}$. You can now build any set you want as $X = \bigcap_{i\notin X}S_i$.

Similarly, an easier proof that a union of recursive sets can be anything at all is to let $T_i=\{i\}$ and now any set&nbsp;$X$ is $X = \bigcup_{i\in X}T_i$.