More About Uniform Upper Bounds on Ideals of Turing Degrees

Harold T. Hodes
The Journal of Symbolic Logic
Vol. 48, No. 2 (Jun., 1983), pp. 441-457
DOI: 10.2307/2273561
Stable URL: http://www.jstor.org/stable/2273561
Page Count: 17

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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: a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump 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 ∪ I = Lα[ A] ∩ ωω for α 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.

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