Gradualist descriptionalist set theory

TL;DR

This paper introduces GDST, a formal set theory with ordinal-indexed interpretations, and explores the properties of a sublanguage NMID related to inductive definitions and reflection principles.

Contribution

It defines a new formal language GDST and establishes equivalences involving reflection properties and well-foundedness of propositions in NMID.

Findings

All propositions in NMID have well-defined truth values iff certain ordinal sequences exist.

Introduces the notion of $ heta$-reflecting ordinals, including admissible and Mahlo types.

Shows the connection between reflection principles and the existence of specific ordinal sequences.

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 $\eta_0 < . . . < \eta_k$ such that for each $i < k$, $\eta_i$ is $\eta_{i+1}$-reflecting, a notion we introduce which implies being $\Pi_n$-reflecting for all $n \in \mathbb N$ (and in particular being admissible and recursively Mahlo).