Quasicreative Sets1
نویسنده
چکیده
The creative sets of Post [4] are the recursively enumerable sets whose complements are productive in the sense of Dekker [l]. In [l], Dekker introduces the class of semiproductive sets, which properly includes the class of productive sets. If we define a set to be semicreative if it is recursively enumerable and has a semiproductive complement, we obtain a natural generalization of creative sets. In this paper, we show that this generalization actually introduces some new sets. We do this by introducing an intermediate class of recursively enumerable sets, and giving an example of a noncreative set in this class. This example also gives two nonseparable recursively enumerable sets which are not effectively nonseparable. We use the notation of [l]. In particular, w„ is the reth recursively enumerable set in a fixed effective enumeration. If « + l=2ai-|• ■ ■ + 2"k with ai> ■ • • >ak, we set a„= {ai, ■ ■ ■ ,ak}. Thus every nonempty finite set is an an lor just one n. A set A is quasiproductive if there is a partial recursive function / such that if con(ZA, then f(n) is defined, afw(ZA, and «/<„)(£ w„. We then call / a quasiproductive function of A. A set is quasicreative if it is recursively enumerable and has a quasiproductive complement. A quasicreative set is semicreative [l, Proposition G]; the author does not know if the converse holds.
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