ABSTRACT
Then A and A c are trivial, and additionally, the latter is a strongly c.e. real. (For instance, if we assume that A is computable, then the stage when the first i bits of A stop moving allows us to define a computable function g , after which o(i) cannot drop below i, contrary to Theorem 52. Also a(i) can drop below i only i (in fact log i) times, and hence A exists.)
We remark that the first construction clearly combines with, for instance, permitting: below any c.e. nonzero degree there is a noncomputable c.e. K - trivial set. Also, one can use it to construct a promptly simple one, or a variant to avoid a given low c.e. set, using the Robinson trick. However, we do not know if there is a complete ^T-trivial set. An answer either way would be very interesting. In the positive way it says that the relationship between <% and <t fails as badly as it can. In the negative way, then the first proof of Theorem 51 above provides a priority-free (or more precisely “injury-free”) solution to Post’s problem.8 While these are known, the construction we give is particularly simple. The Kucera-Terwijn result of the next section is another example of this phenomenom.
