Definiteness properties of first-order schemes

TL;DR

This paper introduces a formal framework for analyzing the definiteness of first-order schemes, refining existing notions and exploring their robustness and distinctions, with implications for foundational schemes like induction and replacement.

Contribution

It proposes the concept of $\

Findings

Introduces $\

concrete examples of schemes separating categoricity notions

Preliminary insights into the definiteness of induction and replacement 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 $\Phi$-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.