What really are the flexible-price limits of a New Keynesian model in the infinitely many firms limit?

Typical New Keynesian models assume that the aggregate price level P can be treated as a given constant in firm optimization problems as the number n of firms goes to infinity. We show that this always holds only when there are actually infinitely many firms and not for the infinitely many firms limit. While largely an irrelevant issue for the flexible-price model, it becomes critical for the flexible-price limit of the sticky price model. Although the flexible-price model has a unique equilibrium, there are infinitely many flexible-price limits of the sticky-price model without interventions even in non-nominal real terms. This does not require dynamic effects, such as the binding zero lower bound. We discuss other defenses of the New Keynesian model assumption and find them either implausible or in need of further discussion due to significant deviations from the conventional analysis.