| 11. | They built a computably enumerable set that is low for Martin-L�f-randomness but not computable.
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| 12. | The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory.
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| 13. | This is not difficult; WKL 0 implies the existence of separating sets for effectively inseparable recursively enumerable sets.
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| 14. | Similarly, the set of logical consequences of any recursively enumerable set of first-order axioms is semidecidable.
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| 15. | The term " enumerable " has the same etymology as in computably enumerable sets of natural numbers.
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| 16. | Some authors require the class " K " to be definable by a recursively enumerable set of presentations.
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| 17. | This is true simply because Diophantine sets, being equal to recursively enumerable sets, are also equal to Turing machines.
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| 18. | This implies in particular that every recursively enumerable set is enumerable by some \ mathcal { E } ^ 0-function.
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| 19. | Any scheme involving a recursively enumerable set of axioms and regular first-order logic will be subject to the incompleteness theorem.
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| 20. | However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.
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