 
Summary: Induction and Inductive Definitions in Fragments
of Second Order Arithmetic
Klaus Aehlig
Abstract
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.
1 Introduction and Related Work
The study of subsystems of second order arithmetic ("Analysis") has a long
tradition in proof theory. Here we investigate a fragment that is defined by a
restriction of the language. By allowing quantification of a second order variable
only for formulae with at most this second order variable free, we obtain a proof
theoretic weaker fragment. This fragment is motivated by a study of Altenkirch
and Coquand [4] who used the nonnested case to obtain a "finitary subsystem
of the polymorphic lambda calculus".
