 … the subsystems for which I have been able to prove the fundamental conjecture are the system with \(\Pi^1_1\) comprehension axiom and a slightly stronger system, that is, the one with \(\Pi^1_1\) comprehension axiom together with inductive definitions.[…] I tried to resolve the fundamental conjecture for the system with the \(\Delta^1_2\) comprehension axiom within our extended version of the finite standpoint. Ultimately, our success was limited to the system with provably \(\Delta^1_2\) comprehension axiom. This was my last successful result in this area. (Takeuti 2003: 133)