 The study of non-standard models did not start with Gödel’s results—Skolem, in particular, was already aware of them earlier in a different context (he had discovered that first-order theories of set theory have unnaturally small, namely, countable models, in Skolem 1922; cf. the entry on  Skolem’s paradox)—but  the first incompleteness theorem elucidates the existence of non-standard models in the context of arithmetic, while the nonstandard models elucidate the first incompleteness theorem. Non-standard models have since then become a rich research area in mathematical logic (see, e.g., Boolos &amp; Jeffrey 1989: Ch. 17; Kaye 1991).