A set X is prime bounding if for every complete atomic
decidable (CAD) theory T there is a prime model 𝔞 of T
decidable in X. It is easy to see that X = 0' is prime bounding.
Denisov claimed that every X ≤T 0' is not prime bounding,
but we discovered this to be incorrect. 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 non-low2 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 φ(\overline{x}) 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.