 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 characterizing finitism and predicativism in mathematically precise way. In more formal terms, the starting point is a theory \(\bT_0\) which is accepted as correct and an extension procedure \(\bT\mapsto\bT'\) which is viewed as leading from a correct theory T to a more encompassing correct theory \(\bT'\). Moreover, the language of these theories is supposed to contain a formula \(\rAcc(x)\) such that provability of \(\rAcc(\bar{a})\) in a correct theory entails that  \(a\in{\cO}\).[23]  Kreisel singled out two autonomous progressions of theories \(\{{{\bF}}_a\}\) and \(\{{{\bR}}_a\}\) for finitism and predicativity, respectively, and determined the least upper bound of the \(\lvert a\rvert\) appearing in the first hierarchy to be the ordinal \(\varepsilon_0\) which is also the proof-theoretic ordinal of PA. The determination of the least upper bound for the predicative hierarchy \(\{{{\bR}}_a\}\) was achieved independently by Feferman (1964) and Schütte (1964, 1965). It turned out that this ordinal can be expressed in a notation system developed by Veblen that will be presented next.