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