Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 899–924 | Cite as

Fragments of Kripke–Platek set theory and the metamathematics of \(\alpha \)-recursion theory

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The foundation scheme in set theory asserts that every nonempty class has an \(\in \)-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and \(\alpha \)-recursion theory. We take KP set theory without foundation (called KP\(^-\)) as the base theory. We show that KP\(^-\) + \(\Pi _1\)-Foundation + \(V=L\) is enough to carry out finite injury arguments in \(\alpha \)-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP.


Metamathematics Foundation Kripke–Platek set theory \(\alpha \)-recursion theory 

Mathematics Subject Classification



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Authors and Affiliations

  1. 1.Kurt Gödel Research Center for Mathematical LogicUniversity of ViennaViennaAustria

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