Definiteness properties of first-order schemes
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 V\"a\"an\"anen, among others. To this end, we introduce the notion of -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.
Keywords
Cite
@article{arxiv.2511.21954,
title = {Definiteness properties of first-order schemes},
author = {Piotr Gruza and Mateusz Łełyk},
journal= {arXiv preprint arXiv:2511.21954},
year = {2025}
}