Journal of Symbolic Logic

More About Uniform Upper Bounds on Ideals of Turing Degrees

Harold T. Hodes

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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: $\mathbf{a}$ 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.

Article information

Source
J. Symbolic Logic, Volume 48, Issue 2 (1983), 441-457.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183741260

Mathematical Reviews number (MathSciNet)
MR704098

Zentralblatt MATH identifier
0514.03027

JSTOR
links.jstor.org

Citation

Hodes, Harold T. More About Uniform Upper Bounds on Ideals of Turing Degrees. J. Symbolic Logic 48 (1983), no. 2, 441--457. https://projecteuclid.org/euclid.jsl/1183741260


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