More About Uniform Upper Bounds on Ideals of Turing Degrees
Hodes, Harold T.
J. Symbolic Logic, Tome 48 (1983) no. 1, p. 441-457 / Harvested from Project Euclid
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.
Publié le : 1983-06-14
Classification: 
@article{1183741260,
     author = {Hodes, Harold T.},
     title = {More About Uniform Upper Bounds on Ideals of Turing Degrees},
     journal = {J. Symbolic Logic},
     volume = {48},
     number = {1},
     year = {1983},
     pages = { 441-457},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1183741260}
}
Hodes, Harold T. More About Uniform Upper Bounds on Ideals of Turing Degrees. J. Symbolic Logic, Tome 48 (1983) no. 1, pp.  441-457. http://gdmltest.u-ga.fr/item/1183741260/