Global and local admissibility. II. Major subsets and automorphisms (Q1060215)

[For part I see Patras Logic Symp., Proc., Patras/Greece 1980, Stud. Logic Found. Math. 109, 325-333 (1982; Zbl 0525.03034).] An r.e. set A is a major subset of an r.e. set B if \(A\subseteq B\) and \(B\setminus A\) is infinite, but there is no r.e. set C such that \(A\subseteq C\subseteq B\) and both \(B\setminus C\) and \(C\setminus A\) are infinite. Major subsets were first introduced in \(\alpha\)-recursion theory as a substitute for maximal sets that often fail to exist. The author studies the case \(\alpha =\aleph_{\omega}\), an example of an effectively singular ordinal. He investigates r.e. sets A that are locally recursive, i.e. \(A\upharpoonright \aleph_ n\) is \(\aleph_ n\)- recursive for sufficiently large n. He strengthens the notation in two steps, uniformly locally recursive and strongly uniformly locally recursive. If an r.e. set is not of the first kind the author shows that it has a major subset. If a non-recursive r.e. set is of the second kind, it has no major subsets. An example will be \(S(\omega)=\{\beta <\alpha |\) of \((\beta)=\omega \}\). He also shows that no r.e. \(D\leq_{\alpha}S(\omega)\) has a major subset. Finally the author gives applications of his results to questions about automorphisms in the lattice of r.e. sets, ordered by inclusion and factorized by the \(\alpha\)-finite sets. He finds incomplete r.e. degrees a and b such that for no pair \(A\in a\), \(B\in b\), there is an automorphism sending A to B.