Maddy’s Notion of ‘Maximize’ and Strong Theories of Sets

This paper examines contemporary approaches to maximization in axiom selection debates in set theory, applying these methods to the current debate between \(V=Ult(L)\) and forcingForcing axioms. The paper first surveys the motivations for maximization as a tool in axiom selection debates, focusing in particular on the formal approach of Maddy. It then examines an important current debate in set-theoretic practiceSet-theoretic practice between strong axiom candidates capable of settling the value of the continuum: the inner modelInner model axiom \(V=Ult(L)\) and strong forcingForcing axioms. Finally, the paper will apply the formal notions of maximizeMaximize to these axiom candidates, showing that–under highly plausible conjectures–forcingForcing axioms strictly maximize over \(V=Ult(L)\) . The paper will then briefly conclude with an examination of the implications of this result for axiom selection as a whole.