Standard analysis

Let $\mathcal{L}$ be a first-order two-sorted language. Let $S$ be some fixed structure. A standard structure is an $\mathcal{L}$-structure of the form $(M,S)$, where $M$ is arbitrary. When $S$ is a compact topological space (and $\mathcal{L}$ meets a few additional requirements) it is possible to adapt a significant part of model theory to the restricted class of standard structures. This has been shown by Henson and Iovino for normed spaces and has been generalized to a larger class of structures in [AAVV]. We further generalize their approach to a significantly larger class. The starting point is to prove that every standard structure has a positive elementary extension that is standard and realizes all positive types that are finitely consistent. A second important step is to prove that (in a sufficiently saturated structure) the negation of a positive formula is equivalent to to an infinite (but small) disjunction of positive formulas. The main tool is the notion of approximation of a positive formula and of its negation that have been introduced by Henson and Iovino. We review and elaborate on the properties of positive formulas and their approximations. In parallel, we introduce continuous formulas which provide a better counterpart to Henson and Iovino theory of normed spaces and to the real-valued model theory of metric spaces [BBHU]. To demonstrate this setting in action we discuss countable categoricity and local stability.