Hierarchical Multiverse of Sets

Abstract: In this paper, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of `degrees of intentionality' of theories. The presented view is articulated for the purpose of reconciling epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favour of a hierarchical picture of the multiverse in which theories or models are thought to be ordered with respect to their plausibility, as a manifestation of endorsing the idea that some set theories are more plausible than others. The proposed multiverse account settles the pluralist's dilemma, the dichotomy that there is a trade-off between the richness of mathematical ontology and the objectivity of mathematical truth. The view also extends and serves as an alternative position to Balaguer's intention-based Platonism from which he claims that a certain version of mathematical pluralism follows.