With this project I propose to investigate the notion of arbitrary object using historical, philosophical and logical methodologies. The privileged perspective from which the analysis will be conducted is that of set theory. The project is divided in three complementary axes. First we propose a general reconstruction of the theorization about arbitrary objects, stressing the role that concepts and their extensions played in shaping axiomatic set theory. Moreover, we plan to reconstruct the origin, and to understand the meaning, of the use of…