Unreducible features in set theory

Abstract

The universalist position in set theory maintains that there is only a single, maximal universe of sets and, as a result, all sentences about these objects are ideally verifiable. Often, those who subscribe to this view are committed to offering a sensible account to alternative universes familiar to many mathematicians. In this article, we will analyze the reduction strategies offered by universalists. Recently, Enayat in [1] proved that no two models of ZF are bi-interpretable, while Hamkins and I in [2] proved that no two well-founded models of
ZF are mutually interpretable. In view of these results, we will argue that the range of the construction for alternative universes in a single universe is limited. Thus, the adherents of an alternative universe have sufficient grounds to reject the alleged copy offered by the universalist as a faithful copy. Finally, we will argue that the reasons for adding new elements to the multiverse should be specific instead of being the result of an emulation in a previously known universe.