Parts of Classes

Lewis's 1991 'Parts of Classes' — mereological reformulation of set theory

Tradition: Analytic metaphysics / mereology / philosophy of mathematics

Lewis's 1991 mereological reformulation of set theory — classes have proper parts, not membership

Published by Blackwell in 1991, 'Parts of Classes' offers Lewis's mereological reformulation of set theory. Lewis argues that the relation between a singleton class and its member is the only set-theoretic primitive that needs to be assumed; every other set-theoretic notion (membership, subset, intersection, union, the empty class) can be defined in terms of singletons plus mereology (the calculus of parts and wholes). The book's central thesis is that classes have proper parts — the singleton of a class is a class that contains other singletons as parts. The technical core (a system of mereology plus a singleton-forming operation) is set out in detail and shown to be equivalent in expressive power to standard ZF set theory, with an appendix by John P. Burgess and A. P. Hazen providing the formal-technical work. The book is a major contribution to the philosophy of mathematics and to the application of mereology beyond ordinary physical objects. Lewis's broader philosophical thesis — that set theory can be reduced to a more philosophically tractable structure (mereology) plus a single primitive (singleton) — connects with his earlier 'On the Plurality of Worlds' (1986) and with his structural realism about mathematics generally.