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Syntactic translations and provably recursive functions. (English) Zbl 0593.03038
In this paper conservation results for classical theories over the corresponding intuitionistic theories are given using Kolmogorov’s version of the so-called double-negation translation (1925). The author provides conditions under which a formula \(\phi\) is provable from a set \(\Gamma\) of formulas in minimal logic (intuitionistic logic resp.) if it is classically. From certain formulas satisfying these conditions he concludes that a recursive function is provably total in so-called intuitionistic full arithmetic, if it is in classical full arithmetic. The same holds for Heyting and Peano Arithmetic and for functionals of higher type instead of recursive functions. If a recursive function is provably total in classical ZF, so it is in intuitionistic ZF. The author introduces another map from formulas to formulas, which translates intuitionistic provability into provability in minimal logic. He establishes conditions for \(\Gamma\) and \(\phi\) such that \(\Gamma\) \(\vdash \neg \neg \phi\) entails \(\Gamma \vdash_ M\phi\) respectively \(\Gamma \vdash_ I\phi\). These results cover some work of H. Friedman on closure of theories under Markov’s and other rules.
As the author stresses by himself, in his paper strong results are obtained by familiar and easy methods, and simple proofs, and - that he does not mention - cleverly chosen concepts.
Reviewer: H.Pfeiffer

03F50 Metamathematics of constructive systems
03D20 Recursive functions and relations, subrecursive hierarchies
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