HIlbert's tenth problem involves Diophantine equations. It involves an obvious truth and asks for a proof - an algorithm. Matiyasevich, among others, proved that it was unprovable. It's an example, or perhaps corollary to the Incompleteness Theorem.
I suspect it involves 'axiomatic' truth, which is theoretical. I further suspect that maybe it is akin to uncertainty principle, which is a limit imposed by the model and not the true nature of the energy 'particle.'
I am asking for an example of something that is true but cannot be proven in the real world, not in a mathematical model.