2016 | 25 | 3 Mereology and Beyond (II) | 285-308
Article title

Set-theoretic mereology

Title variants
Languages of publication
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.
Physical description
  • 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,
  • Graduate School of System Informatics, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan,
  • A. Baudisch, D. Seese, P. Tuschik, and M. Weese, “Decidability and quantifier-elimination”, pages 235–270 in Model-theoretic logics, Perspect. Math. Logic, Springer, New York, 1985.
  • L. Champollion and M. Krifka, “Mereology”, in Cambridge Handbook of Semantics, P. Dekker and M. Aloni (eds.) Cambridge University Press (in press).
  • C.C. Chang and H.J. Keisler, Model theory, volume 73 of “Studies in Logic and the Foundations of Mathematics”, North-Holland Publishing Co., Amsterdam, third edition, 1990.
  • Ju.L. Eršov, “Decidability of the elementary theory of relatively complemented lattices and of the theory of filters”, Algebra i Logika Sem., 3 3 (1964): 17–38.
  • J.D. Hamkins, “Is the inclusion version of Kunen inconsistency theorem true?” MathOverflow answer, 2013. (accessed 25.04.2016).
  • G. Hellman, “Mereology in philosophy of mathematics”, preprint available on the author’s web page at
  • W. Hodges, Model Theory, volume 42 of “Encyclopedia of Mathematics and its Applications”, Cambridge University Press, Cambridge, 1993.
  • A. Kanamori, “The empty set, the singleton, and the ordered pair”, Bull. Symbolic Logic, 9, 3 (2003): 273–298. DOI:10.2178/bsl/1058448674
  • D. Lewis, Parts of Classes, Blackwell, 1991.
  • J.D. Monk, Mathematical Logic, Springer-Verlag, New York–Heidelberg, 1976. Graduate Texts in Mathematics, No. 37.
  • B. Poizat, A Course in Model Theory, Universitext, Springer-Verlag, New York, 2000. An introduction to contemporary mathematical logic, Translated from the French by Moses Klein and revised by the author.
  • A. Varzi, “Mereology”, in The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), E.N. Zalta (ed.).
  • M. Weese, “Decidable extensions of the theory of Boolean algebras”, pages 983–1066 in Handbook of Boolean algebras, Vol. 3, North-Holland, Amsterdam, 1989.
Document Type
Publication order reference
YADDA identifier
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.