In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.

If α is a singular cardinal (either real or fake) in L, I exhibit many natural α-re subsets, defined uniformly from the ▵ 1 subsets of α. If α is a true cardinal this provides an uppersemilattice (usl) embedding from the lattice of ▵ 1 subsets of α into the usl of α-re-degrees. It will also be shown that this embedding cannot be extended to the Σ 1 subsets of α.

Bailey, C. and R. Downey, Tabular degrees in \Ga-recursion theory, Annals of Pure and Applied Logic 55 205–236. We introduce several generalizations of the truth-table and weak-truth-table reducibilities to \Ga-recursion theory. A number of examples are given of theorems that lift from \Gw-recursion theory, and of theorems that do not. In particular it is shown that the regular sets theorem fails and that not all natural generalizations of wtt are the same.

I consider the projectum of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\beta$\end{document}-r.e. set. It is shown that there are tame r.e. sets with small projecta and that there are tame r.e. sets with large projecta.