Mereology and Set Theory
Maybe someone here can explain what the hell this is supposed to mean.
http://en.wikipedia.org/wiki/Mereology
I'm not seeing the difference here. Mereology seems to be a language to grasp partially ordered sets. What makes it different from set theory then?
From the same wiki article, not the best exposition to my mind but should give you a flavour of what sets them apart. If you want further info let me know and I'll try and ellucidate further.
From
http://en.wikipedia.org/wiki/Mereology
[edit] Mereology and set theory
Stanisław Leśniewski rejected set theory, a stance that has come to be known as nominalism. For a long time, nearly all philosophers and mathematicians avoided mereology, seeing it as tantamount to a rejection of set theory. 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 1941.
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. 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.
Many logicians and philosophers reject these motivations, on such grounds as:
They deny that sets are in any way ontologically suspect;
Occam's Razor, when applied to abstract objects like sets, is either a dubious principle or simply false;
Mereology itself is guilty of proliferating new and ontologically suspect entities such as fusions.
For a survey of attempts to found mathematics without using set theory, see Burgess and Rosen (1997).
In the 1970s, thanks in part to Eberle (1970), it gradually came to be understood that one can employ mereology regardless of one's ontological stance regarding sets. This understanding is called the "ontological innocence" of mereology. This innocence stems from the mereology being formalizable in either of two equivalent ways:
Quantified variables ranging over a universe of sets;
Schematic predicates with a single free variable.
Once it became clear that mereology is not tantamount to a denial of set theory, mereology became largely accepted as a useful tool for formal ontology and metaphysics.
In set theory, singletons are "atoms" which have no (non-empty) proper parts; many consider set theory useless or incoherent (not "well-founded") if sets cannot be built up from unit sets. The calculus of individuals was thought to require that an object either have no proper parts, in which case it is an "atom", or to be the mereological sum of atoms. Eberle (1970) showed how to construct a calculus of individuals lacking "atoms", i.e., one where every object has a "proper part" (defined below) so that the universe is infinite.
There are analogies between the axioms of mereology and those of standard Zermelo-Fraenkel set theory (ZF), if Parthood is taken as analogous to subset in set theory. On the relation of mereology and ZF, also see Bunt (1985). One of the very few contemporary set theorist to discuss mereology is Potter (2004).
Lewis (1991) went further, showing informally that mereology, augmented by a few ontological assumptions and plural quantification, and some novel reasoning about singletons, yields a system in which a given individual can be both a member and a subset of another individual. In the resulting system, the axioms of ZFC (and of Peano arithmetic) are theorems.
Forrest (2002) revises Lewis's analysis by first formulating a generalization of CEM, called "Heyting mereology", whose sole nonlogical primitive is Proper Part, assumed transitive and antireflexive. There exists a "fictitious" null individual that is a proper part of every individual. Two schemas assert that every lattice join exists (lattices are complete) and that meet distributes over join. On this Heyting mereology Forrest erects a theory of pseudosets, adequate for all purposes to which sets have been put.
