Bounding prime models

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 nonlow2 X is indeed prime bounding, we fix a function f ≤T X which dominates every total 0′-computable function g. Given a CAD theory T , we use f to eventually find, for every formula φ(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.