Fragments of Kripke-Platek set theory and the metamathematics of \(\alpha \)-recursion theory (Q334991)

In the paper under review, \(\mathrm{KP}^-\) (\(\mathrm{KP}\) without foundation) with restricted foundation is investigated. A number of basic results were proved. For example, it is shown that \(\mathrm{KP}^-\) + \(\Pi_1\)-foundation implies \(\Sigma_1\)-\(\epsilon\)-induction, and \(\mathrm{KP}^-\) + \(\Pi_{n+1}\)-foundation + \(\Sigma_{n+1}\)-collection + \(V = L\) implies \(\Sigma_{n+1}\)-foundation for all \(n\in \mathbb{N}\). Then, the authors apply the results to investigate \(\alpha\)-recursion theory. It is shown that both the Friedber-Muchnik theorem and Sacks splitting theorem hold under \(\mathrm{KP}^-\) + \(\Pi_1\)-foundation + \(V=L\).