Set theory as a foundation for mathematics - universism and multiversism

Set theory has performed since its formulation – and moreover, its axiomatization by Zermelo – the role of being a language common to all mathematics. The idea of unifying all mathematics, as well as being a maximal model under which all sentences of mathematics can be dealt with seemed a very fitting concept of what the purposes of set theory should be. However, the independence results stroke set-theorists with the possibility of new approaches to the unifying role assumed by set theory. By means of forcing new set theories could be conceived and thus the maximal universe that hitherto served as the foundation for all mathematics could be extended. Should set theorists then resort to assume there are many set theories instead of one? According to Maddy [Maddy, 2017], the universalist view satisfies many of the conditions needed for a theory to be considered a foundational theory for mathematics, to wit, ‘Shared Standard’ – provides a standard way of carrying proofs in mathematics –, ‘Generous Arena’ – is the theory in which every interaction between mathematical objects occur – and ‘Metamathematical Corral’ – is the theory that allows mathematicians to carry out metamathe-matical investigations – whereas the multiversist, despite also satisfying these conditions, do not do so in the best possible way. For Maddy, the main problem with the multiversist view is that it cannot satisfy ’Generous Arena’ in the sense that the universist view does. In turn, Ternullo [Ternullo, 2019] defends that the multiversism not only satisfies every condition posed by Maddy, but does so in a satisfatory way, which better comes in terms with what has been developed in set theory contemporarily. My goal is to compare Maddy’s and Ternullo’s argument and, assuming  that set theory is the best foundational theory for mathematics, give criteria that try to determine the question: which view should we adopt: the universist or the multiversist?

References

[Maddy, 2017] Maddy, P. (2017). Set-theoretic foundations. In Andrés Eduardo Caicedo, James Cummings, P. K. and Larson, P. B., editors, Foundations of Mathematics, volume 690 of Contemporary Mathematics, pages 289–322. American Mathematical Society.

[Ternullo, 2019] Ternullo, C. (2019). Maddy on the multiverse. In Stefania Centrone, Deborah Kant, D. S., editor, Reflections on the Foundations of Mathematics - Univalent Foundations, Set Theory and General Thoughts, volume 407 of Studies in Epistemology, Logic, Methodology,and Philosophy of Science, pages 43–78. Springer, 1st edition.