Berkeley Cardinals and the search for V

This thesis is concerned with Berkeley Cardinals, very large cardinal axioms inconsistent with the Axiom of Choice. These notions have been recently introduced by J. Bagaria, P. Koellner and W. H. Woodin; our aim is to provide an introductory account of their features and of the motivations for investigating their consequences. As a noteworthy advance in the topic, we establish the independence from ZF of the cofinality of the least Berkeley cardinal, which is indeed the main point to focus on when dealing with the failure of Choice. We then explore the structural properties of the inner model L(V_\delta+1) under the assumption that delta is a singular limit of Berkeley cardinals each of which is a limit of extendible cardinals, lifting some of the theory of the axiom I_0 to the level of Berkeley cardinals. Finally, we discuss the role of Berkeley cardinals within the ultimate project of attaining a "definitive" description of the universe of set theory.