The Journal of Symbolic Logic

Research Article

Induction and inductive definitions in fragments of second order arithmetic

Klaus Aehlig

Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany,, E-mail:


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 definition.

(Received May 24 2004)

(Revised February 17 2005)