Let A be the axiomatic system whose only axiom is the sentence n $\approx$ n and deduction rules $\varphi \rightarrow [\varphi \vee \psi ], \psi \rightarrow [\varphi \vee \psi ], (\varphi, \psi) \rightarrow [\varphi \wedge \psi ]$.
Prove that this system is valid, effective but not complete and that the set of all the theorems of A is primitive recursive.
Do not know how to prove that the system is valid, would have to look if each rule of deduction applied the axiom is a theorem and this is enough? to prove the incompleteness I create a truth that is not a theorem how to do this. Thanks