Double Recursion Theorems

16 Chapter 0. Prerequisitesthe provable formulas of S is an arithmetic set. Since every r.e.set is obviously arithmetic, then every axiomatizable system is alsoan arithmetic system. And now we have the following result of AlfredTarski:Theorem 9. The system N is not even arithmetic let alone axiomatizable.Since N is complete, Theorem 9 follows from the following result:Theorem 9". Every arithmetic subsystem of N is incomplete.Proof of Th. 9". Suppose