Stable formulas in standard structures

Continuous logic, introduced by Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov in "Model theory for metric structures" (2008), has replaced the logic of Henson-Iovino, presented in "Ultraproducts in analysis" (2002), as a formalism for studying the model theory of continuous structures. One of the first articles on the subject, "Continuous first order logic and local stability" by Itaï Ben Yaacov and Alexander Usvyatsov (2010), defines local stability within continuous logic—an aspect that, as noted by Henson, was missing from the approach in "Ultraproducts in analysis". Subsequently, this has been used as a major argument in favor of Model theory for metric structures over Henson and Iovino's approach. Recently, a classical approach to continuous structures has been proposed by Claudio Agostini, Stefano Baratella, Silvia Barbina, Luca Motto Ros, and Domenico Zambella in "Continuous logic in a classical setting" (2024) and by Domenico Zambella in "Standard analysis" (2023), which properly extends the class of structures covered by the framework. These articles introduced the notion of a structure with a standard sort. The goal of this thesis is to extend the notion of stable formula to these structures and to prove the analogue of one of the most important properties of stable formulas: every set externally definable by a stable formula is definable.