31. The set of all provable sentences in an effective axiomatic system is always a recursively enumerable set. 32. As intermediate results, Post defined natural types of recursively enumerable sets like the hypersimple and hyperhypersimple sets. 33. From the A3000 design onwards, it was deemed desirable for all enumerable hardware expansions to use Autoconfig. 34. Every recursively enumerable ( or even hyperarithmetic ) nonempty subset of this total ordering has a least element. 35. :A language is recursively enumerable if there is a Turing machine that can generate all its strings. 36. They built a computably enumerable set that is low for Martin-L�f-randomness but not computable. 37. Equivalently, a set is recursively enumerable if and only if it is the range of some computable function. 38. The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory. 39. The study of arbitrary ( not necessarily recursively enumerable ) Turing degrees involves the study of the Turing jump. 40. This is not difficult; WKL 0 implies the existence of separating sets for effectively inseparable recursively enumerable sets.
