Just want to point out that your answer is essentially incorrect. Godel's completeness theorem has nothing to do with the existence of models. It is about the provability of sentences that are true within all models.

Also, the incompleteness theorem doesn't say anything about the nonexistence of a model. It gives the existence of at least one sentence X (for sufficiently nice theories that include enough arithmetic) such that there ARE two models where X is true in one and false in the other.