Locally Computable Structures

We introduce the notion of a locally computable structure, a natural way of generalizing the notions of computable model theory to uncountable structures ${\mathcal{S}}$ by presenting the finitely generated substructures of $\S$ effectively. Our discussion emphasizes definitions and examples, but does prove two significant results. First, our notion of m-extensional local computability of ${\mathcal{S}}$ ensures that the Σ n -theory of ${\mathcal{S}}$ will be Σ n for all n≤ m+ 1. Second, our notion of perfect local computability is equivalent (for countable structures) to the classic definition of computable presentability.