Let $P_0$ be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let $M$ be a countable, nonstandard model of $P_0$ whose domain we suppose to be the standard integers. Let $T$ be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment $M^'$ of $M$ which is a model of $T$ such that the complete diagram of $M^'$ is Turing reducible to the atomic diagram of $M$. Moreover, neither the addition nor the multiplication of $M$ is recursive.