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On the optimality of conservation results for local reflection in arithmetic. (English) Zbl 1316.03036

The paper answers a question asked by L. D. Beklemishev in “Notes on local reflection principles” [Theoria 63, 139–146 (1997)]. It is shown that \(\Pi_{2}\)-sentences are not conserved for T = EA + “\(f\) is total” where EA denotes elementary arithmetic and \(f\) is any nondecreasing computable function with elementary graph. A generalization of this result to \(n>0\) is also considered and it is proved that for \(n > 0\), \(I \Pi_{n+1}^{-}\) is conservative over \(I \Sigma_{n}\) with respect to \(\Pi_{n+2}\)-sentences.

MSC:

03F30 First-order arithmetic and fragments
03D20 Recursive functions and relations, subrecursive hierarchies
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