Axiomatic Set Theory and Unincreasable Infinity

In this paper we propose a reflection on the use of axiomatic set theory as a fundamental tool to address the foundational issues of mathematics. In particular, we focus on the key concept of infinty, indeed the strong"the Absolute" by Cantor), as we aim to show how the point of view offered by a specific set-theoretical framework allows us to deal with such a paradoxical notion in a completely safe manner. For this purpose, we shall introduce NBG set theory and discuss its consistency. We assume the reader is familiar with ZFC set theory (see for example [2] or [3]).