Let $L$ be a language and $\bar{L}$ be its complement. Which one of the following is NOT a viable possibility?

A fundamental theorem in computability theory states that a language $L$ is recursive (decidable) if and only if both $L$ and its complement $\bar{L}$ are recursively enumerable (r.e.).

From this theorem, the possible relationships between $L$ and $\bar{L}$ are:

1. Both $L$ and $\bar{L}$ are recursive (and hence both are r.e.).

2. One of $L$ or $\bar{L}$ is r.e. but not recursive; the other must NOT be r.e.

3. Neither $L$ nor $\bar{L}$ is r.e.

It is impossible for both $L$ and $\bar{L}$ to be r.e. without both being recursive. Therefore, option C is NOT a viable possibility.