Set-theoretic mereology

Joel David Hamkins; Makoto Kikuchi

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arXiv · 1601.06593

Set-theoretic mereology

Joel David Hamkins·Makoto Kikuchi

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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.

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Joel David Hamkins, Makoto Kikuchi. 2016-04-25. Set-theoretic mereology. https://arxiv.org/abs/1601.06593

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