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Induction and inductive definitions in fragments of second order arithmetic
 The Journal of Symbolic Logic
"... 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 var ..."
Abstract

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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
Certifying
"... polynomial time and linear/polynomial space for imperative programs ..."
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