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## Re: [ontolog-forum] Ontological Assumptions of FOL

 To: Pat Hayes "[ontolog-forum]" "John F. Sowa" Sun, 18 Mar 2007 09:25:56 -0500 <45FD4BF4.3020406@xxxxxxxxxxx>
 ```Pat,    (01) The point I was trying to make is that there are an open-ended number of possible representations of anything, including the universe. And I don't believe that any particular method is ideal for all purposes for all time.    (02) JFS>> Many people prefer to use mereology instead of set theory.    (03) PH> Use for what, John?    (04) As a formalism for representing claims about some domain.    (05) PH> ... if we go around saying that the universe "is" a > mereological sum, we are in exactly the same danger.    (06) Of course. I would never make either claim.    (07) The critical distinction between set theory and mereology is that the axioms of set theory generate something new:    (08) For any x, the set {x} is distinct from x.    (09) But the formalism of mereology makes no such assumption:    (010) For any x, the sum of all the parts of x is identical to x.    (011) A typical example is to consider the old way of dividing the territory of France into provinces and the newer way of dividing it into departments:    (012) 1. Using sets, the set of all provinces, the set of all departments, and the territory of France are treated as three distinct entities (or things or whatever you want to call them).    (013) 2. Using mereology, the sum of all provinces, the sum of all departments, and the territory of France are one and the same.    (014) PH> Not all things have parts (what is a part of the number 17?)    (015) I would prefer to use some symbol instead of the word "part" because the common word "part" is used in many different ways. But whatever you want to call it, the fundamental axiom of mereology is that everything (including the number 17) is a "part" (or whatever) of itself.    (016) PH> And set theory does support virtually all of mathematics, > whereas mereology supports virtually none.    (017) That is a critical distinction, but it is not a criticism. It simply means that set theory, by itself, can generate new entities (or things), but mereology cannot. Therefore, an additional generative axiom must be coupled with mereology in order to build up new structures.    (018) For example, set theory has two operators, memberOf and subsetOf, but mereology has only one operator, partOf. Both subsetOf and partOf obey the following axioms:    (019) 1. Every set/sum is a subset/part of itself.    (020) 2. The subset/partOf relation is a partial ordering.    (021) The memberOf relation is Cantor's most significant innovation. That is what gives set theory its generative power:    (022) For all x, there exists a y such that memberOf(x,y).    (023) In particular, the simplest such y is {x}. To start, all you need is the empty set {}, and you can build your way up to infinities of infinities.    (024) I admire Cantor's magnificent creation, but I don't see any reason to prefer it. If you use mereology as your theory of collections, you need to add some additional generative operator. My preference is Peano's successor function: start with 0, and build up the integers as s(0), s(s(0)), s(s(s(0))), etc.    (025) Starting with mereology plus s(), you can build up all of arithmetic, including Dedekind cuts for the reals.    (026) For geometry, Whitehead chose mereology (actually, he invented his own version) for the planned fourth volume of the Principia. His primitive elements were blobs of 4D space-time, but I don't remember what generative axioms he used.    (027) Tarski independently built up a theory of solid geometry based on mereology, in which the only primitive elements were spheres. He defined a point as a "fictitious" limit of a descending sequence of nested spheres. Lines and planes are also fictitious limits. Then he showed that Euclid's axioms could be stated in terms of those fictitious things. Tarski's approach is actually a more realistic model of physical solids than Euclid's.    (028) I personally prefer the Peano-Whitehead-Tarski approach of assuming domain-dependent generative mechanisms instead of building up all of mathematics by repeatedly nesting sets of sets of empty sets. All those nested empty sets are impressive, but not very perspicuous.    (029) John    (030) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (031) ```
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