Lewis on Set Theory
David Lewis in the short monograph Parts of Classes (Lewis 1991, henceforth PC) undertakes a fundamental re-examination of the relationship between merelogy, the general theory of parts, and set theory, the general theory of collections. He assumes a certain minimum background familiarity with both subjects, and limitations of space make the same assumption inevitable in the present account. Varzi (2011) is recommended as a clear, concise survey of mereology in general. Karen Bennett's chapter in the present volume discusses Lewis's version of mereology in particular. Among the asumptions surveyed by Varzi, Lewis's are about the strongest, including unrestricted composition, the claim that any things whatsoever have a fusion, but leaving open whether everything is a fusion of atoms, or things with no parts other than themselves. For set theory in the form in which it will be considered here, Boolos (1989) (with Boolos (1971) in the background), to which reference will in any case be essential, should suffice.
Lewis became aware of the possibility of certain technical improvements too late in the production process for his book to incorporate them except as an appendix to PC (Burgess & al 1991, henceforth PC*), co-authored with A. P. Hazen and the present writer and including also a contribution by W. V. Quine. Lewis became aware of the possibility of certain further technical improvements too late to incorporate them in the book at all, whence the follow-up paper, "Mathematics is Megethology" (Lewis 1993, henceforth MM), which Lewis described in its introductory session as "an abridgement of parts of [PC] not as it is, but as it would have been had I known sooner what I know now."
A half-dozen themes pursued in the book and article: (1) formulation of theses on how the mereological notion of part applies to classes; (2) restatement of traditional mereologists' complaints about set theory, concentrating on the notion of singleton sets, and motivating a "structuralist" approach to be further explained below; (3) defense beyond what is already found in (Lewis 1986) of the background assumptions of mereology, including controversial theses on "ontological innocence" and "composition as identity"; (4) elaboration of a framework combining mereology with the plural quantification of Boolos (1984 and 1985); (5) consideration of how to simulate within such a framework, using assumptions related to the axiom of choice, quantification over relations; (6) development of the structuralist treatment of set theory using such simulated quantification, showing that beyond the framework the only assumptions needed are about how many atoms there are, whence the identification of mathematics with "megethology," the theory of size.
I will take up the six themes in the order listed. The most important sources are as follows: PC chapter 1 for (1); PC chapter 2 for (2); PC §3.6 for (3); PC chapter 3 for (4); PC* plus Hazen (1997) for (5); and MM plus Boolos (1989) for (6).