Direct constuction of nonlow noncomplete c.e. sets How can one construct a noncomplete nonlow c.e. set?
(Background: I've been trying to construct, as an exercise, a nonlow low$_2$ set, but I do not know what kind of requirement is adequate for nonlowness, and I thought a simple example would help.  I searched Odifreddi Vol 2, Soare  but in vain.  I know how to indirectly achieve nonlowness, e.g., by making it high, but that's not applicable to the exercise.)
A pointer to a construction or the construction itself will be appreciated.
 A: Jump inversion is your friend here, and gets you all the way to the problem you mention.
The Sacks jump inversion theorem (there are other jump inversion theorems, the Friebderg and the Shoenfield being the most common) states that for any $x\ge_T0'$ which is c.e. in $0'$, there is some c.e. $y$ such that $y'\equiv_Tx$. The proof is an infinite injury argument, but definitely one of the easier ones.
Meanwhile, any of the usual constructions of a low noncomputable set (Friedberg-Muchnik, low simple set, etc.) relativize to produce a low-over-$0'$ set - that is, an $x$ which is c.e. in $0'$, $>_T0'$, and such that $x'\equiv_T0''$.
Combining these two facts immediately yields a low$_2$, nonlow c.e. degree. Note that we can easily modify this argument to get a low$_3$ c.e. degree which is nonlow$_2$, etc.

By the way, for a really sweet example of jump inversion combined with the recursion theorem, check out Sacks' shiny little box: http://www.ams.org/journals/proc/1967-018-01/S0002-9939-1967-0207558-7/S0002-9939-1967-0207558-7.pdf. Sacks gives an elementary(ish) proof of a theorem originally due to Lachlan and Martin, that there is a c.e. set which is neither high$_n$ nor low$_n$ for any $n\in\omega$.
