The language of the theories contains a formula Acc(x) such that provability of Acc(a-bar) in a correct theory entails that a belongs to the constructive ordinals O

In the foregoing progressions the ordinals remained external to the theory. Autonomous progressions of theories are the proper internalization of the general concept of progressions. In the autonomous case one is allowed to ascend to a theory \(\bT_a\) only if one already has shown in a previously accepted theory \(\bT_b\) that \(a\in{\cO}\). This idea of generating a hierarchy of theories via a boot-strapping process appeared for the first time in Kreisel 1960, where it was proposed as a way of