 In the 1960’s, Tait (1960, 1965, 2010) extended Ackermann’s methods to obtain an ordinal analysis of extensions of arithmetic with principles of transfinite induction. More streamlined and modern versions of this approach can be found in Mints 2001 and Avigad 2002b. More recently, Mints, Tupailo, and Buchholz have considered stronger, yet still predicatively justifiable, fragments of analysis, including theories of arithmetic comprehension and a \(\Delta^{1}_1\)-comprehension rule (Mints, Tupailo &amp; Buchholz 1996; Mints &amp; Tupailo 1999; see also Mints 2016). Arai 2002 has extended the epsilon substitution method to theories that allow one to iterate arithmetic comprehension along primitive recursive well orderings. In particular, his work yields ordinal analyses for predicative fragments of analysis involving transfinite hierarchies and transfinite induction.