The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make (...) it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. (shrink)

We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...) dependence of certain versions on (highly independent) cardinal arithmetic. (shrink)

Alternative models of idealized scientific inquiry are investigated and compared. Particular attention is devoted to paradigms in which a scientist is required to determine the truth of a given sentence in the structure giving rise to his data.