Saul Kripke gave a proof of incompleteness using nonstandard models and a notion of "fulfillability". Roughly speaking, a sequence fulfills a (prenex) formula if the formula is true when its successive quantifiers are bounded by the terms of the sequence. Kripke apparently never published this, but Hilary Putnam presented in a lecture, subsequently published as "Nonstandard models and Kripke's proof of the Gödel theorem" [Notre Dame Journal of Formal Logic 41 (2000) pp. 53-58]. The MathSciNet review by Alex Wilkie reads:
"This paper was developed from a lecture given by the author at Beijing University in 1984 which described Kripke's notion of fulfillability and how it may be used to give a proof of the incompleteness of Peano arithmetic. Putnam has decided to publish it now because Kripke has still not published it himself. The point of this proof is that it is semantic and avoids self-reference, but is much simpler than the Paris-Harrington argument. It achieves this simplicity by considering a sentence that directly expresses the existence of long (finite) sequences of natural numbers that verify all the bounded approximations of the Peano Axioms, rather than deducing this existence, as Paris and Harrington had to, from a version of Ramsey's Theorem. Thus the simplicity comes at the cost of mathematical naturality. Also, although it is not made explicit in this paper, one still has to go through the process of Gödel numbering and the construction of a uniform satisfaction predicate for bounded quantifier formulas."