Due to Godel's incompleteness theorems we know that there are true statements in a system that cannot be proven with that system. My questions are 1) can we tell which statements in a system are the ones that cannot be proven, and 2) can we choose through the design of the system which statements will not be provable?
The motivation for my questions is simply the observation of how many results depend on the Riemann conjecture and how important to mathematics it is, along with the fact that it is almost universally believed to be true, yet it resists proof. Similar things can be said about the twin prime conjecture and many other unproven conjectures. Therefore, I ask the questions: is is possible that the Riemann conjecture (or twin prime conjecture) is one of those statements that is true but can never be proven? And can we know whether it is one of those questions?
If first order Peano arithmetic is consistent, there is no algorithm that will determine, given input $\varphi$, whether or not $\varphi$ is neither provable nor refutable from first-order PA.
At our current state of knowledge, it is possible that the twin prime conjecture is true but not provable in first order PA.