This question is related to Godel's incompleteness theorem, which states that no sufficiently complex formal system can be both consistent and complete.
Is it possible to add axioms to a formal system $P$ to generate a new formal system $P'$ such that $P'$ is consistent and all unprovable statements in $P$ have proof in $P'$? (The way that this would circumvent Godel's incompleteness theorem is that there would be new unprovable statements in $P'$ that weren't in $P$).
For example, primitive recursive arithmetic is a formal system in which everything is provable, however it can only express a subset of the theories that can be expressed by Peano arithmatic. In the same way, $P'$ would be more expressive than $P$, and primitive recursive arithmatic represents a trivial case of $P$ where the above question is true. I am interested in $P$ where there exist unprovable statements.
Adding more expressivity is not going to help you if Gödel's incompleteness theorem applies to $P$.
One of the premises of Gödel's theorem is that the language of the theory can express a certain amount of arithmetic sentences (level $\Pi^0_1$ in the arithmetical hierarchy, if I recall correctly).
If we add more expressivity to $P$, producing $P'$ with a finite number of consistent additional axioms, then Gödel's theorem still applies to $P'$. But the $P'$-undecidable sentence that the theorem produces will be one of the arithmetic sentences that already $P$ could express.
So $P'$ will have an undecidable sentence that was in the language of $P$.