Löwenheim–Skolem theorems and nonstandard models

Sometimes we specify a structure by giving a description and counting anything that satisfies the description as just another model of it. But at other times we start from a conception we try to articulate, and then our articulation may fail to pin down what we had in mind. Sets seem to have had such a fate. For millennia sets lay fallow in logic, but when cultivated by mathematics in the nineteenth century, they seemed to bear both a foundation and a theory of the infinite. The paradoxes of set theory seemed to threaten this promise. With an eye to proving freedom from paradox, versions of set theory were articulated rigorously. But around 1920, Löwenheim and Skolem proved that no such formalized set theory can come out true only in the hugely infinite world it seemed to reveal, for if it is true in such a world, it will also be true in a world of the smallest infinite size. (Versions of this remain true even if we augment the standard expressive devices used to formalize set theory.) But then, Skolem inferred, we cannot articulate sets determinately enough for them to constitute a firm foundation for mathematics.