Source: J. Symbolic Logic
Volume 70, Issue 4
A fragment with the same provably recursive functions as n
iterated inductive definitions is obtained by restricting second order
arithmetic in the following way. The underlying language allows
only up to n+1 nested second order quantifications and those
are in such a way, that no second order variable occurs
free in the scope of another second order quantifier.
The amount of induction on arithmetical formulae only affects the
arithmetical consequences of these theories, whereas adding induction
for arbitrary formulae increases the strength by one inductive
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