```Date: Feb 6, 2013 6:14 AM
Author: Zaljohar@gmail.com
Subject: Sets as Memory traces.

Suppose that we have three bricks, A,B,C, now one can understand theWhole of those bricks to be an object that have every part of itoverlapping with brick A or B or C, lets denote that whole by W. Ofcourse clearly W is not a brick, W is the totality of all the threeabove mentioned bricks. However here I want to capture the idea of'membership' of that whole, more specifically what do we mean when wesay that brick A is a 'member' of W. We know that A is a part of W,but being a part of W is not enough by itself to qualify A as being amember of W, one can observe that brick A itself can have many properparts of it and those would be parts of W of course (since part-hoodis transitive) and yet non of those is a member of W. So for a part ofW to be a member of W there must be some property that it mustsatisfy. I'll add another primitive binary relation in addition topart-hood, and that binary relation I'll call as "contact". So we'llbe working within a kind of Mereotopology. However axioms tocharacterize contact relation would be different from those ofMereotopology. Here only disjoint (non overlapping) objects areallowed to be in contact. When we say A is in contact with B then wemean that for some x,y where x is a part of A and y is a part of Bthere do not exist a gap between x and y, of course as said aboveprovided that A and B are disjoint objects.Now we come to stipulate the sufficient condition for membership of awhole, this is:x is a member of y iff x part of y And (there do not exist a part of ythat is in contact with x) And every proper part of x is in contactwith some part of x.This supplies us with the impression that x is a 'solid' entity andyet x is separate (not in contact) from some other parts of y shouldthose exist.However this interpretation of membership using this kind ofmereotopology has its shortcomings, the greatest is that it islimiting in the sense that only one level of membership is possible,that is between solid blocks and collections of them, any collectionof several blocks would not be able to be a member of any object sinceit does have separate proper parts. So this would only be enough tointerpret flat sets.If we desire having a hierarchy of sets being members of other setsand if we want also to keep the above background of thinking ofmatters in terms of parts and wholes and contact etc.., then we need amore complex approach, one of those would be to invoke TIME in thepicture!This without doubt would complicate the whole picture but yet it doessupply us with some interpretation of sets and their membership.Now instead of having a binary relation C to represent contact, weupgrade that to a triple relation symbol C(x,y,t) to signify x incontact with y at moment t.This would revolutionize how matters are tackled here. So for exampleif at moment t1 we have a whole apple P being a solid block i.e. it isnot in contact with any object and all its proper parts in contactwith some proper part of it, then P would be said to be a block at t1.However this doesn't mean that apple P would also remain in this blockstatus, possibly at moment t2 the same apple P had been cut into twoseparate halfs, so at moment t2 P is a whole of two separate blocks P1and P2 and no longer being as a solid block, so at moment t2 P cannotbe an element of any object, while at moment t1 P could have been.This development would introduce us to the concept of MEMORY traces!and of course the introduction of a new binary primitive 'memorizedin' or 'carved in'.A memory trace is supposed to be a record of objects in block status.So for example suppose that an rock was immersed in some mud at momentt1 and thus left its print on that piece of mud, then after a whilethat rock was broken into two smaller parts, and one part also felldown on another part of the mud and made another print on that pieceof mud, so suppose that mud remained like that for years, now thispiece of mud have the prints of the whole rock at solid status andalso of a part of that rock at solid status, this piece of mud wouldbe considered as a 'memory trace'.Now we would coin another interpretation of 'set' as a 'memory trace'.Membership would be of objects in solid status carved in the memorytrace.so x carved in y  or x memorized in y is taken to mean that x is asolid block at some moment t and memorized as such in y.So we have the axiom.x M y -> Exist t. x is solid at t.So membership can be interpreted as this memorizing relation and setscan be interpreted as 'memory traces'It is natural to assume identity of memory traces after what ismemorized in them.It needs to be stressed that memory traces are NOT the wholes of whatis memorized in them! since the whole of an apple and a half of it isthe apple itself and it is not different from the whole of threethirds of it, but the memory traces of those are different!However if an object do not change its solid status over time, i.e. ifwe have the following property:for all t. x is solid at tthen wholes (i.e. totalities) of such objects can be taken to bememory traces of them since time is not having any differential effecton those kinds of objects.So all in all, sets here can be interpreted as memory traces and setmembership as memorizing objects in solid block status.Zuhair
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