Applications of Logic to Analysis

From the beginning of the emergence of new logic, fundamental links have been established between logic and various branches of mathematics, which led to solving mathematical problems and, conversely, solving basic problems in logic itself. One of the challenges of the logical methods in the study of mathematical structures is the impossibility of studying some of the important structures of mathematics, including analytic structures, in the framework of the first-order language and logic. The main purpose of this paper is to provide a suitable logic for studying these structures and then solving problems in the analysis using logical tools. At the beginning of this article, we will briefly review some suitable logics for studying the structures in mathematical analysis, and will outline some of the most important uses of logic in analysis. Then we present and prove one of the recent achievements, which is an important application of logic in analysis. In particular, we study the concept of definability in logic and its relation with mathematical analysis.