The degree of a \(\Sigma_ n\) cut
A reverse-recursion-theoretic investigation of how much strength is needed to prove the existence of minimal degrees or minimal pairs. Take any model of \(P^-+B\Sigma_ n+\neg I\Sigma_ n\) (n\(\geq 2)\), where \(P^-\) \(=\) Peano axioms without induction, \(B\Sigma_ n=\Sigma_ n\)- collection, and \(I\Sigma_ n=\Sigma_ n\)-induction. There exists a \(\Sigma_ n\)-definable cut in this model; let I be any such cut. The authors show that I and \(\emptyset^{n-1}\) form a minimal pair. They likewise show that, under appropriate coding assumptions, such a model also possesses a minimal degree.
scientific article; zbMATH DE number 4172967
scientific article