Journal of Symbolic Logic

Weakly Semirecursive Sets

Carl G. Jockusch, Jr. and James C. Owings, Jr.

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We introduce the notion of "semi-r.e." for subsets of $\omega$, a generalization of "semirecursive" and of "r.e.", and the notion of "weakly semirecursive", a generalization of "semi-r.e.". We show that $A$ is weakly semirecursive iff, for any $n$ numbers $x_1,\ldots,x_n$, knowing how many of these numbers belong to $A$ is equivalent to knowing which of these numbers belong to $A$. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion "$A$ is weakly semirecursive and recursive in $K$" in terms of recursive approximations to $A$. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised.

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J. Symbolic Logic, Volume 55, Issue 2 (1990), 637-644.

First available in Project Euclid: 6 July 2007

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Jockusch, Carl G.; Owings, James C. Weakly Semirecursive Sets. J. Symbolic Logic 55 (1990), no. 2, 637--644.

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