BERKELEY CARDINALS AND THE STRUCTURE OF L(Vδ+1)

Abstract

AbstractWe explore the structural properties of the inner model L(Vδ+1) under the assumption that δ is a singular limit of Berkeley cardinals each of which is itself limit of extendible cardinals, lifting some of the main results of the theory of the axiom I0 to the level of Berkeley cardinals, the strongest known large cardinal axioms. Berkeley cardinals have been recently introduced in [1] and contradict the Axiom of Choice.1 In fact, our background theory will be ZF.