$(\mathcal L, n)$-Models are sequences of finite models approximating a model of an arithmetic theory as formalized in PA. These sequences were first introduced by Shelah and recently studied by the speaker. They allow one to turn questions about consistency and completeness into finitary, combinatorial statements in a straightforward manner, thus giving a relatively uniform and systematic approach to "mathematical" independence in PA. In this talk we will outline their general theory as well as some applications. These include Shelah's example of a true but unprovable $\Pi^0_1$ statement and our recently discovered variation.