In this paper we consider classes whose elements are re- cursively enumerable sets of non-negative integers. No discussion of recur- sively enumerable sets can avoid the use of such classes, so that it seems de- sirable to know some of their properties. We give our attention here to the properties of complete recursive enumerability and complete recursiveness (which may be intuitively interpreted as decidability). Perhaps our most interesting result (and the one which gives this paper its name) is the fact that no nontrivial class is completely recursive. We assume familiarity with a paper of Kleene (2), and with ideas which are well summarized in the first sections of a paper of Post .
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