Definiteness properties of first-order schemes
Piotr Gruza, Mateusz Łełyk
TL;DR
The paper develops Φ-definiteness as a formal, parameterized framework to study scheme definiteness and internal categoricity in foundational theories. It constructs a robust metatheoretical apparatus (schemes, predicative comprehension, strong models, interpretations) to analyze when scheme-level statements are determinate, categorically unique, or complete, across different metatheories and languages. Through separations and preservation results, it demonstrates that categoricity and completeness properties can diverge depending on parameters, cardinalities, and additions like full comprehension, with concrete results for PA and ZF. The work clarifies how definiteness of foundational schemes depends on the surrounding metatheory, offering insights for philosophy of mathematics and the formal study of determinateness in arithmetic and set theory.
Abstract
The paper aims to establish a convenient formal framework for investigating the phenomenon of scheme definiteness, exemplified by first-order internal categoricity as studied by Väänänen, among others. To this end, we introduce the notion of $Φ$-definiteness, thereby refining and extending the conceptual landscape that underlies various first-order categoricity notions in the literature (internal categoricity, strong internal categoricity, intolerance). We provide arguments for the robustness of our definition and present examples of schemes that separate different categoricity- and completeness-like notions. Finally, we offer a brief glimpse into the issue of the definiteness of two canonical foundational schemes - the induction scheme and the replacement scheme.