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Because Sum and Product are binary operations, M6 and M7 admit the sum and product of only a finite number of objects. The fusion axiom, M8, enables taking the sum of infinitely many objects.

The same holds for Product, when defined. At this point, mereology often invokes set theorybut any recourse to set theory is eliminable by replacing a formula with a quantified variable ranging over a universe of sets by a schematic formula with one free variable. The formula comes out true is mereology and the sciences whenever the name of an object that would be a member of mereology and the sciences set if it existed replaces the free variable.

Hence any axiom with sets can be replaced by an axiom schema with monadic atomic subformulae.

M8 and M8' are schemas of just this sort. The syntax of a first-order theory can describe only a denumerable number of sets; hence, only mereology and the sciences many sets may be eliminated in this fashion, but this limitation is not binding for the sort of mathematics contemplated here.

If M8 holds, then W exists for infinite universes.

## Mereology - Wikipedia

Hence, Top need be assumed only if the universe is infinite and M8 does not hold. Top postulating W is not controversial, but Bottom postulating N is.

Hence, while the universe is closed under sum, the product of objects that do not overlap is typically undefined. A system with W but not N is isomorphic to: A Boolean algebra lacking a 0 A join semilattice bounded from above by 1.

Binary fusion mereology and the sciences W interpret join and 1, respectively.

Postulating N renders all possible products definable, but also transforms classical mereology and the sciences mereology into a set-free model of Boolean algebra. If sets are admitted, M8 asserts the existence of the fusion of all members of any nonempty set.

Any mereological system in which M8 holds is called general, and its name includes G. In any general mereology, M6 and M7 are provable. Adding M8 to an extensional mereology results in general extensional mereology, abbreviated GEM; moreover, the extensionality renders the fusion mereology and the sciences.

On the converse, however, if the fusion asserted by M8 is assumed unique, so that M8' replaces M8, then—as Tarski had shown—M3 and M8' suffice to axiomatize GEM, a remarkably economical result. M2 and a finite universe necessarily imply Atomicity, namely that everything either is an atom or includes atoms among its proper parts.

If the universe is infinite, Atomicity requires M9. Adding M9 to any mereological system, X results in the atomistic variant thereof, denoted AX.

### Mereology and the Sciences: Parts and Wholes in the Contemporary Scientific Context

Atomicity permits economies, for instance, assuming that M5' implies Atomicity and extensionality, and yields an alternative axiomatization of AGEM. For a mereology and the sciences time, nearly all philosophers and mathematicians avoided mereology, seeing it as tantamount to a rejection of set theory[ citation needed ].

Goodman too was a nominalist, and his fellow nominalist Richard Milton Martin employed a version of the calculus of individuals throughout his career, starting in Much early work on mereology was motivated by a suspicion that set theory was ontologically suspect, and that Occam's razor requires that one minimise the number of posits in one's theory of the world and of mathematics[ citation needed ].

Mereology replaces talk of "sets" of objects with talk of "sums" of objects, objects being no more than the various things that make up wholes[ citation needed ]. Many logicians and philosophers[ mereology and the sciences