Berkeley Cardinals and the search for V

Raffaella Cutolo

Università di Napoli "Federico II"

The talk will focus on Berkeley cardinals, the strongest known large cardinal axioms, recently introduced by J. Bagaria, P. Koellner and W. H. Woodin. Berkeley cardinals are inconsistent with the Axiom of Choice; their definition is indeed formulated in the context of ZF (Zermelo-Fraenkel set theory without AC). Aim of the talk is to provide an account of their main features and the foundational issues involved. A noteworthy contribution to the topic is my result establishing the independence from ZF of the cofinality of the least Berkeley cardinal, which is in fact connected with the failure of AC; I will describe the forcing notion employed and give a sketch of the proof. In order to show that interesting mathematical consequences can be developed from Berkeley cardinals, I’ll then analyze 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 itself limit of extendible cardinals, lifting some of the theory of the large cardinal axiom I0 to a more general level. Finally, I will discuss the role of Berkeley cardinals within Woodin’s ultimate project of attaining a “definitive” description of the universe of set theory.