More About Uniform Upper Bounds on Ideals of Turing Degrees

Abstract

Let $I$ be a countable jump ideal in $\mathscr{D} = \langle \text{The Turing degrees}, \leq\rangle$. The central theorem of this paper is: $\mathbf{a}$ is a uniform upper bound on $I$ iff $\mathbf{a}$ computes the join of an $I$-exact pair whose double jump $\mathbf{a}^{(1)}$ computes. We may replace "the join of an $I$-exact pair" in the above theorem by "a weak uniform upper bound on $I$". We also answer two minimality questions: the class of uniform upper bounds on $I$ never has a minimal member; if $\cup I = L_\alpha\lbrack A\rbrack \cap ^\omega\omega$ for $\alpha$ admissible or a limit of admissibles, the same holds for nice uniform upper bounds. The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces.