Gradualist descriptionalist set theory

Abstract

We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each $k\in \mathbb{N}$ of a sequence of ordinals $\hat{I}{\cdot }_0<...<\hat{I}{\cdot }_k$ such that for each $i<k$, $\hat{I}{\cdot }_i$ is $\hat{I}{\cdot }_{i+1}$-reflecting, a notion we introduce which implies being ${\hat{I}}_n$-reflecting for all $n\in \mathbb{N}$ (and in particular being admissible and recursively Mahlo).