Abstract: The foundational role of set theory is often tied to a universe view: since set theory provides the foundation of mathematics, it must in principle be able to answer all mathematical questions, including those requiring new axioms beyond ZFC. Against this background, set-theoretic pluralism seems to threaten the foundational role—if there are multiple legitimate extensions of ZFC, some mathematical questions may lack determinate answers.
In this talk, I challenge the assumption that the foundational role of set theory is necessarily linked to the universe view. My argument draws on two key observations about mathematical practice. First, set theory is a relatively isolated field: contemporary research rarely interacts with other areas of mathematics, with exceptions such as Farah’s result on the Calkin algebra. Second, many pluralist set theorists nonetheless regard set theory as foundational. Building on the first observation, their position rests on a compelling point: mathematics outside set theory provides sufficient justification for the ZFC axioms, but not for any axioms beyond ZFC.
This yields a coherent justification of set-theoretic pluralism: if set theory must provide a foundation for mathematics, and ZFC fulfills this role, then set-theoretic pluralism is well grounded. The universist, in turn, faces two options: either justify the universe view on purely set-theoretic grounds, rather than through its foundational role, or demonstrate that new axioms are needed in mathematics to the same extent as, for example, the Axiom of Choice.