A hierarchy for the plus cupping Turing degrees

Journal of Symbolic Logic 68 (3):972-988 (2003)
We say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c.e. degree x with $0 < x \leq a$ , there is a c. e. degree $y \not= 0'$ such that $x \vee y = 0/\'$ . We say that a is n-plus-cupping. if for every c. e. degree x, if $0 < x \leq a$ , then there is a $low_n$ c. e. degree 1 such that $x \vee l = 0'$ . Let PC and $PC_n$ be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then $PC_{1} \subseteq PC_{2} \subseteq PC_3 = PC$ . In this paper we show that $PC_{1} \subset PC_2$ , so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [14] showing that $LC_{1} \subset LC_2$ , as well as extending the Harrington plus-cupping theorem [8]
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DOI 10.2178/jsl/1058448450
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