Results in model theory for Riesz-valued structures

Abstract

We study model-theoretic properties of a logic whose formulas take values in suitable Riesz spaces. In addition to having a set of truth values which is not linearly ordered, there is no absolute truth or falsehood value. We take inspiration from the version of continuous logic developed in [8] by Ben-Yaacov et al. and from the more general approach recently proposed in [11] by Keisler. We extend to our framework a number of results obtained by the above mentioned authors. Under suitable assumptions on the underlying Riesz space, we provide an ultraproduct construction and prove a Łoś theorem, from which compactness follows. In the framework of Riesz spaces we also address definability issues and we extend metric notions, obtaining analogues of Keisler’s pre-metric structures and pre-metric expansions of theories defined in Keisler. In particular, we prove that every theory in a countable language has a pre-metric expansion with a so-called pseudometric approximate distance.