Help me please to come up with an example of two arithmetic formulae $\varphi$ and $\psi$ such that $PA\vdash\varphi\vee\psi$, but neither $\varphi$ nor $\psi$ is derivable in $PA$ ($PA$ is Peano arithmetics)
Upd: I made a mistake, so, formulae must not have any free varibles
This is equivalent to showing that PA is an incomplete theory, which is a consequence of Godel's incompleteness theorem. That is, by the theorem, there exists $\varphi$ such that neither $\varphi$ nor $\neg\varphi$ is derivable from PA. Then just take $\psi$ to be $\neg\varphi$. (I should add that this obviously assumes that PA is consistent)