Playing with admissibility spectra (Q1111551)

Let A be a countable sequence of countable \(\Sigma_ n(A)\)-admissible ordinals. Jensen's result on representations of A-admissible sequences A is strengthened to the following: for every \(m<n\) there is a set of natural numbers R such that for every \(\alpha <\sup A:\) (1) \(\alpha\) is \(\Sigma_ n(R)\)-admissible iff \(\alpha\) is in A; (2) if \(\alpha\) is \(\Sigma_ n(A)\)-admissible and is not in A then \(\alpha\) is \(\Sigma_ m(R)\)-non-projectible and not \(\Sigma_{m+1}(R)\)-admissible; (3) if \(k<n\) and \(\alpha\) is \(\Sigma_ k(A)\)-admissible but not \(\Sigma_ n(A)\)-admisible then \(\alpha\) is \(\Sigma_ k(R)\)-admissible. Also there is an R such that (1) and (3) hold plus: (2a) if \(\alpha\) is \(\Sigma_ n(A)\)-admissible and is not in A then \(\alpha\) is \(\Sigma_ m(R)\)- admissible and not \(\Sigma_ m(R)\)-non-projectible.