 And (35) is the closest we can get to saying that x is composed of the φs, i.e., all and only those entities that satisfy the given condition φ (in the present case: being an atomic part of x): every φ is part of x, and any part of x overlaps some φ. Indeed, provided the φs are pairwise disjoint, this is the standard definition of what it means for something x to be composed of the φs (van Inwagen 1990: 29), and surely enough, if the φs are all atomic, then they are pairwise disjoint. Thus, although (P.7) does not say that everything is ultimately composed of atoms, it implies it—at least in the presence of (P.1) and (P.2). (Of course, non-standard mereologies in which either postulates is rejected may not warrant the initial equivalence, so in such theories (33) would perhaps be a better way to express the assumption of atomism.) In another way, however, (34) may still not be enough. For if the domain is infinite, (P.7) admits of models that seem to run afoul of the atomistic doctrine. A simple example is a descending chain of decomposition that never “bottoms out”, as in Figure 6: here x is ultimately composed of atoms, but the pattern of decomposition that goes down the right branch “looks” awfully similar to a gunky precipice. For a concrete example (from Eberle 1970: 75), consider the set of all subsets of the natural numbers, with parthood modeled by the subset relation. In such a universe, each singleton {n} will count as an atom and each infinite set {m: m &gt; n} will be “made up” of atoms. Yet the set of all such infinite sets will be infinitely descending. Models of this sort do not violate the idea that everything is ultimately composed of atoms. However, they violate the idea that everything can be decomposed into its ultimate constituents. And this may be found problematic if atomism is meant to carry the weight of metaphysical grounding: as J. Schaffer puts it, the atomist's ontology seems to drain away “down a bottomless pit” (2007: 184); being is “infinitely deferred, never achieved” (2010: 62). Are there any ways available to the atomist to avoid this charge? One option would simply be to require that every model be finite, or that it involve only a finite set of atoms. Yet such requirements, besides being philosophically harsh and controversial even among atomists, cannot be formally implemented in first-order mereology, the former for well-known model-theoretic reasons and the latter in view of the above-mentioned result by Hodges and Lewis (1968). The only reasonable option would seem to be a genuine strengthening of Atomicity in the spirit of what Cotnoir (2013c) calls “superatomism”. Given any object x, (P.7) guarantees the existence of some parthood chain that bottoms out at an atom. Superatomicity would require that every parthood chain of x bottoms out—a property that fails in the model of Figure 6. At the moment, such ways of strengthening (P.7) have not been explored. However, in view of the connection between classical mereology and Boolean algebras (see below, Section 4.4), mathematical models for superatomistic mereologies may be recovered from the work on superatomic Boolean algebras initiated by Mostowski and Tarski (1939) and eventually systematized in Day (1960). (A Boolean algebra is superatomic if and only if every subalgebra is atomic, as with the algebra generated by the finite subsets of a given set; see Day 1967 for an overview.) See also Shiver (2015) for ways of strengthening (P.7) in the context of stronger mereologies such as GEM (Section 4.4), or within theories formulated in languages enriched with set variables or plural quantification.