The onto-logical fundamentals of Alain Badiou's metaphysics: the relation between being and being-there

With this paper we try to outline the essential onto-logic foundation of the Alain Badiou's metaphysical proposal; and also identify the indispensable dialectical links to elaborate what we can consider a contemporary metaphysics of the infinite multiple-against the classical dogmatic metaphysics of the One and the philosophies of the finitude-which incorporates in a dialectically inclusive way mathematics of set theory, in Being and event, and mathematical logic of category theory, in Logics of Worlds. Without going in depth into the details of necessary key concepts to think truth procedures in Badiou's philosophy, we will see how dialectical relations that are established inside this new metaphysics at an ontological level concentrate all the historicity of the thought of classical correlations, i.e. being/appearing, or being/being-there, univocity/equivocity, object/relation, extrinsic/intrinsic, binary logic/transcendental logic and mathematics/logic, in the philosophical thought of the contemporary dialectical relation between set theory and category theory.