Lewis on Set Theory David Lewis in the short monograph

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§1. Parts of Classes

The notion of "part" is applied in ordinary language to entities of many different sorts. As the introductory portion of Varzi (2011) makes plain, however, the word is not, in ordinary language, always applied in the same sense. To claim logic-like universality for mereology — such a claim is not fully explicit in Lewis, though his thought tends in that direction — is to claim there is a core sense of "part" that can be applied univocally to entities of any sort whatsoever, including classes, if such there be. It is the applicability of the notion of and the mereological theory of "parthood" to sets and classes that is Lewis's most fundamental claim.

The form of the theory of sets and classes of concern to Lewis is a versions of what is variously called "second-order Zermelo-Frankel set theory with choice" (henceforth second-order ZFC) or "Morse-Kelly set theory" (henceforth MK) to be met with in the literature. Both versions can be developed to as to admit individuals (German: Urelemente), items that are neither sets nor classes but can belong to such collections. In the version followed by Lewis, that of the lumpers as opposed to the splitters, sets are classes of a kind, the "small" ones and equivalently the ones that can be members of other classes; those of the other kind, the "large" ones and equivalently the ones that cannot be members of other classes, are called proper classes.

Lewis has one main thesis concerning mereology and classes, with one outstanding corollary. His main thesis follows immediately from two subordinate theses. The second of these is deduced from three yet further subordinate theses.

Main Thesis: The parts of a class are precisely its subclasses.

First Thesis: One class is part of another if and only if it is a subclass.

Second Thesis: Any part of a class is a class.

Division Thesis: There are only individuals, classes, and fusions thereof.

Priority Thesis: No class is a part of any individual.

Fusion Thesis: Any fusion of individuals is an individual.

Corollary: Singletons are atoms.
Besides a deduction of the Second Thesis from Division and Priority and Fusion, Lewis offers various motivating considerations and heuristic arguments in favor of these theses, but the justification of some ultimately remains largely pragmatic: The assumptions together lead to a powerful and attractive theory.

It is a consequence of these theses and mereology that the principle of the universality of set theory, according to which any condition determines a class whose members are all only those things for which the condition holds, bar proper classes, must fail, at least assuming for nontriviality that there is at least one individual and therefore at least one class. For that principle implies a strengthened version of division according to which there are only individuals and classes, whereas the fusion of an individual with a class cannot be a class by the Second Thesis, since it has an individual as a part, and cannot be an individual by the Priority Thesis, since it has a class as a part. Thus mereology and set theory cannot both be universal, and Lewis opts for mereology.

In practice, Lewis generally ignores fusions of individuals and classes, in effect tacitly assuming one is quantifying only over individuals and classes. For the most part, he also in effect assumes one is quantifying only over atoms (including singletons) and fusions thereof (including classes), to the exclusion of what he calls atomless gunk, if such there be. He sketches an adaptation of his results the case where there is gunk. (If all there is is gunk, the assumption that there are lots of nonoverlapping globs of it will do in place of the assumption that there are lots of atoms in the overall construction.)

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