Here we give the correct characterization that the prime bounding
sets
X ≤T 0' are exactly the sets which are not
low2. Recall that
X is low2 if X'' ≤T 0''. To
prove that a low2 set X is not prime bounding
we use a 0'-computable listing of the array of sets {Y :
Y ≤T X} to build a CAD theory T
which diagonalizes against all potential
X-decidable prime models A of T. To prove that
any nonlow2 X is indeed prime bounding, we
fix a function f ≤T X that is not
dominated by a certain 0'-computable function that picks out
generators of principal types. Given a CAD theory T, we use
f to eventually find, for every formula consistent with
T, a principal type which contains it, and hence to build an
X-decidable prime model of T. We prove the prime
bounding property equivalent to several other combinatorial
properties, including some related to the limitwise monotonic
functions which have been introduced elsewhere in computable model
theory.