Elements of set theory and recursive arguments

In this paper we provide a self-contained introduction to some of the basic topics of Mathematical Analysis, comprising natural and unrestricted set theoretic methods. The note reflects partially the contents of a lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018. More precisely, following a quite new didactic approach, we recall here some basic facts on the Generalized Induction Principle as well as the Recursion Theorem, which plays a crucial role in the foundation of Mathematical Logic. Some elements of von Neumann, Gödel and Bernays (NGB) set theory are given in the last section. The note provides the preliminary tools that are essential in order to study the classical notion of Dedekind completeness