Classes of recursively enumerable sets and their decision problems
TLDR
This paper considers classes whose elements are re-cursively enumerable sets of non-negative integers whose properties are complete recursive enumerability and complete recursiveness.Abstract:
1. Introduction. 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 desirable 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 [5](2), and with ideas which are well summarized in the first sections of a paper of Post Í7].read more
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References
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Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I
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An Unsolvable Problem of Elementary Number Theory
TL;DR: The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world as mentioned in this paper, which is supported by libraries, scholarly societies, publishers, and foundations.
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Recursively enumerable sets of positive integers and their decision problems
TL;DR: The notion of recursive functions of positive integers has been studied in the context of symbolic logic as discussed by the authors, and it has been shown that such a concept admits of development into a mathematical theory much as the group concept has been developed into a theory of groups.
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On notation for ordinal numbers
TL;DR: In this paper, the authors consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties: given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of a given ordinal.
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