Mereology

Mereology is the theory of the part–whole relation and of derived operations such as the mereological sum. (The sum of several things is the smallest thing of which they are all parts.) It was introduced by Leśniewski to avoid Russell’s paradox.

Unlike the set-membership relation, the part–whole relation is transitive. This makes mereology much weaker than set theory, but gives the advantage of ontological parsimony. For example, mereology does not posit the proliferation of entities found in set theory, such as $\mathrm{\varnothing },\{\mathrm{\varnothing }\},\{\{\mathrm{\varnothing }\}\},\dots .$

Mereology has occasioned controversy: over whether many things really have a mereological sum if they are either scattered or, even worse, of different categories; over the uniqueness of sums; and over Lewis’ claim that the non-empty subsets of a set are literally parts of it.