Comment by pdonis

> I don’t think incompleteness is best described as saying "no first-order axioms can pin down a single model"

Well, it's an obvious implication of the two theorems (completeness and incompleteness) combined: if a FO sentence is not provable from a system of FO axioms, it can't be true in all models of those axioms. And if an FO sentence is not provable, its negation can't be either (since proving its negation true would prove the sentence itself false). So the negation also can't be true in all models. That means there must be at least one model in which the FO sentence is false, and at least one in which its negation is false (so the sentence itself is true). Which means there must be at least two models of that set of FO axioms--i.e., the axioms can't pin down a single model. And the incompleteness theorem proves that there is such a sentence in every system of FO axioms.

I agree that the Lowenheim-Skolem theorem has similar consequences, since it says there must be models with different infinite cardinalities.