Noah has given an excellent answer. Here is another way to look at such an answer.
Theorem. For EVERY computably enumerable set $R$ with non-computable complement $R'$ and for every computably axiomatizable consistent theory $T$, there is a Turing machine that enumerates $R$, such that $T$ does not prove any assertion of the form $n\in R'$, using that enumeration of $R$.
Proof. Fix any enumeration of $R$, and then modify it to produce a Turing machine that also enumerates everything into $R$, if a proof of a contradiction from $T$ is found. Since $T$ does not prove that $T$ has no such proof, $T$ cannot prove that any assertion of the form $n\in R'$. But meanwhile, since $T$ really is consistent, then this program will enumerate $R$ correctly. QED
Thus, the property you are asking about is not a property of the predicate or the c.e. set you have in mind, but rather it is a property of your way of describing how that set is enumerated. In other words, what you have is an intensional property of the set rather than an extensional one.