Set-theoretic mereology

TL;DR

This paper investigates a set-theoretic approach to mereology based on inclusion, establishing its limitations as a foundation for mathematics while highlighting the robustness of augmented versions.

Contribution

It formalizes the axioms of inclusion-based mereology, proving its limitations as a standalone foundation and exploring enhancements like the singleton operator.

Findings

Inclusion-based mereology is finitely axiomatizable and decidable.

It cannot define the element relation rom inclusion alone.

Augmented mereology with singleton operators is more foundationally robust.

Abstract

We consider a set-theoretic version of mereology based on the inclusion relation $\subseteq$ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of $\in$ from $\subseteq$, we identify the natural axioms for $\subseteq$-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.