Firstly, the incompleteness theorem (there's only one) is not about computation but about proofs; specifically that there are some things within an axiomatic system which are true which cannot be proven such within the system itself.
Secondly, this does not mean that these things are unprovable, period. For instance, Gödel showed that you can prove these things perfectly fine, you just have to use different system. But this system will have its own unprovable statements.
Thirdly, computability was Turing's domain, not Gödels, and has to do with the kinds of algorithms you can use to perform proofs, not the system the proof is for. For example: second-order logic (SOL) can prove things about first-order logic (FOL) that FOL cannot prove about itself, but both systems can by utilized by a universal turing machine.
Fourthly, there is no known thing that cannot be proven, period, which is obvious for us humans.