Model theoretic proofs of the incompleteness theorems

Godel's proofs of the incompleteness theorems are frequently discussed in the context of constructive or nitary viewpoint. Godel showed a concrete independent statement of arithmetic and the proof does not contain any transcendental argument. On the other hand, the completeness theorem asserts that a sentence is not derivable from a theory if and only if there exists a model of the theory which does not satisfy the sentence, and it suggests the existence of model-theoretic proofs of the incompleteness theorems. Nevertheless, as Kreisel states, \as far as, e.g., constructive independence proofs are concerned models are, by the nature of the case, useless." ([Kr2], page 381.)