 The goal of giving an ordinal analysis of full second order arithmetic has not been attained yet. For many years \(\Pi^1_2\)-comprehension posed a formidable challenge and the quest for its ordinal analysis attained something of a holy grail status (cf. Feferman 1989). At first blush it might be difficult to see why the latter comprehension is so much more powerful than \(\Delta^1_2\)-comprehension (plus \({{\BI}}\)). To get a sense for the enormous difference, it seems advisable to work in (admissible) set theory and consider a hierarchy of recursively large ordinal notions wherein these comprehension schemes correspond to the bottom and the top end of the scale, respectively. That is discussed in  Appendix D.  Here we just mention a few reductions to “constructive” frameworks. The reductions we have in mind, underlie a broadened view of “constructivity”. Constructive theories of functions and sets that relate to Bishop’s constructive mathematics as theories like ZFC relate to Cantorian set theory have been proposed by Myhill, Martin–Löf, Feferman and Aczel. Among those are Feferman’s constructive theory of operations and classes, \(\bT_0\) in 1975 and 1979, Martin-Löf’s intuitionistic type theory (1984) and constructive set theory, especially Constructive Zermelo-Fraenkel Set Theory, CZF, the latter also combined with the regular extension axiom, REA. By employing an ordinal analysis for set theory KPi which is an extension of Kripke-Platek set theory via an axiom asserting that every set is contained in an admissible set (see  Appendix D)  it has been shown that KPi and consequently \((\Delta^1_2{\Hy}\bCA)+{{\BI}}\) can be reduced to both of these theories: