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We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-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.
- Mathematics, Philosophy, Computer Science, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, NY 10016 & Mathematics, College of Staten Island of CUNY, Staten Island, NY 10314, http://jdh.hamkins.org
- Graduate School of System Informatics, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan, http://www2.kobe-u.ac.jp/~mkikuchi/index-e.html
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