As discussed in this MathOverflow question, I'm trying to find what the result would be of applying a Feferman-Scutte-like analysis to the predicativism of Edward Nelson and Charles Parsons, who believe that mathematical induction is impredicative. The first step of such an analysis was done in a paper by Burgess and Hazen.
Basically, we start out with Robinson's $Q$, and then we add (ramified) predicative second-order logic, which is like regular second-order logic except that the comprehension schema is broken up into levels. The comprehension schema for level $0$ sets only allows formulas with no quantification over sets. The schema for level $1$ sets allows quantification over level $0$ sets. For any natural number $n$, the schema for level $n+1$ sets allows quantification over sets of level $n$ and below. Now Burgess and Hazen essentially showed that predicative second-order Robinson arithmetic implies exponential function arithmetic, also known as $EFA$. (Actually, they showed that it implies Robinson arithmetic + induction for formulas with bounded quantifiers + the totality of exponentiation, but I'm pretty sure $EFA$ is a conservative extension of that.)
But there's no particular reason why we should stop at finite levels. For instance, we could have a comprehension schema for level $\omega$ sets, which allows quantification over sets of any finite level. And so on, for bigger and bigger transfinite ordinals. The question is just what ordinals do we use? Following Feferman-Schutte, we only allow an ordinal $\alpha$ if we can show that it is "predicatively acceptable" (using lower-level comprehension schemata).
So how do we begin? For starters, as discussed above predicative second-order Robinson arithmetic with finite levels implies $EFA$, and the proof theoretic ordinal of EFA is $\omega^3$, so we can say that the ordinals less than $\omega^3$ are predicatively acceptable.
So now the question becomes, if we take $EFA$ and we add predicative second-order logic with comprehension schemes for levels up to $\omega^3$, how much of first-order arithmetic can we prove? This would help in making a dent in my MathOverflow question.
Any help would be greatly appreciated.
Thank You in Advance.
EDIT: Someone mentioned in responding to my MathOverflow question that this paper by Leivant shows that "predicative stratification in the polymorphic lambda calculus using levels $<\omega^\ell$ leads to definability of functions in Grzegorczyk's $\mathscr E_{\ell+4}$". I'm not that familiar with the lambda calculus, so can someone confirm that this implies that $EFA$ with predicative second-order logic with comprehension schemes for levels up to $<\omega^3$ proves that all functions in Grzegorczyk's $\mathscr E_{7}$ are total? If that were true then the proof-theoretic ordinal of this system would be $\omega^7$, so it would be a first step in doing a Feferman-Schutte-like analysis. (And by similar methods, I think we can go to $\omega^{11}$, $\omega^{15}$, etc, all the way up to $\omega^\omega$, the proof-theoretic ordinal of $PRA$.)
