A theory T is tight if no two distinct theories extending T are bi-interpretable. A theory T is semantically tight if no two non-isomorphic models of T are bi-interpretable. Visser has proved that PA is tight. Enayat followed by showing that ZF is tight. In this talk, we explore the philosophical significance of tightness with regards to the issue of monism and pluralism in mathematics. Does tightness support the one view over the other? Or is it a neutral property in this debate? We shall argue that although tight theories have features that are pleasing to both monists and pluralists, the ultimate answers to the questions above depend on the role of the theories in question. In particular, we shall argue that PA and ZF are distinguished on this issue by the foundational role of the latter. As extensions of ZF are expected to play a foundational role in mathematics, it matters that whatever extension we adopt (or whatever model we consider as the intended one) can adequately interpret other interesting mathematical theories (or structures). The tightness of ZF, we argue, limits this possibility and thus paves the way for pluralism about set theory.
