Recusively Enumerable or Recursive dependent on whether P=NP If a language is defined such that
$L = (0+1)^{\ast}$   if $\mathsf{P} = \mathsf{NP}$ and $\emptyset$   otherwise

Then $L$ is a regular language if $\mathsf{P} = \mathsf{NP}$, otherwise it is the empty langauge. 
Therefore $\mathsf{P} = \mathsf{NP}$ , $L$ is recursive (being regular), but is $L$ still recursive if $\mathsf{P} \neq \mathsf{NP}$?
The language $L = \emptyset$  is indeed a recursive set.