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 generic objects in the more abstract fields of contemporary mathematical practice. Later, from a more philosophical perspective, we plan to to confront the concept of arbitrary set with the notions of sets based either on Gödel’s iterative conception or on Cantor’s principle of limitation of size. From a more intensional perspective on the theory of sets, we plan to study whether arbitrary objects may be considered as resulting from a process of abstraction à la Frege, by proposing a philosophy of mathematical language inspired by Searle’s theory of speech acts. Finally we plan to analyze more formally the notion of arbitrariness by means of that of genericity, on which the technique of forcing rests. We propose to axiomatize genericity in an abstract setting. Moreover we intent to apply modal logic (RI-logics) in order to capture what is invariant under forcing (Ω-logic), in the attempt to obtain insights in the solution of the Ω-conjecture. Finally we propose to use forcing itself in order to study the relative or absolute character of the notion of genericity.