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Imaging temporal Mereotopology
Posted:
Apr 3, 2013 4:41 PM


In my last explanation about sets, one thing that I must mention is that the size criterion followed in standard set theories is not clearly motivated by it. There is no clear relationship between the size of a collection and the ability to have a label (name) for it? We can change the metaphor of "label" or "name" to that of "image", then this can be motivated since some collections might be so big that they cannot be imaged! This would fit most well founded sets and it would break down with non well foundedness. However there is another line of imagination that follows Temporal Mereotopology, in which another primitive is added to parthood and identity, that is "timed Contact".
Timed contact is a Three place relation symbol, the notation of it is the following
C(x,y,t) standing for x in contact with y at moment t.
No being in "contact" mean that the object in contact to each other are disjoint (discrete) i.e. no overlap between them (share no common part), and that there is no gap between some parts of them. Of course all of that relative to some moment t, so if x and y are in contact with each other at t they might not be so at another moment t*.
Axioms for C(x,y,t) are:
Ax.1: C(x,y,t) > C(y,x,t) Ax.2: C(x,y,t) & x,y are not atoms >Exist a,b: a PP x & b PP y & C(a,b,t) Ax.3: (For all z. C(x,z,t) <>C(y,z,t)) > x=y Ax.4: (Exist t. C(x,y,t)) > ~ x O y
Now a very important concept is that of a temporal mereotopological unit.
notation: Unit(x,t) read as x is a unit at moment t. The informal idea is that for x to be compact. This is defined as:
Unit(x,t) iff [~ Exist y. C(x,y,t)] & [For all q. q PP x > Exist g. g PP x & C(q,g,t)]
Unitof(x,y,t) read as: x is a unit of y at moment t. This is defined as:
Unitof(x,y,t) iff Unit(x,t) & x P y
Now we come to add another primitive relation that is "image", the idea is that of copying but the difference is that a copy will never break down into further smaller Unites at any moment t. Also the idea of this methodology is guarantee having Images of Units. Of course images of two units might come into contact with each other at some moment t at then they'd become a unit and they'd be guaranteed an image!
Ax5: x Image of y > ~Exist z,t. z PP x & Unit(z,t) Ax6: x image of y & k image of q > ~ x O k Ax.7. Exist t. Unit(x,t) > Exist m. m image of x & Unit(m,t)
The nice idea is to define Membership after the following relation:
x E y iff Class(y) & Exist z. z P y & z image of x
Class would be defined as the collection of All images of some units.
Now to understand that: lets say we have two unites A,B at some moment t, then those would be guaranteed to have images A*,A2*,..B1*, B2* ...that are unites at moment t. Now the collection of All those images is the set {A,b}. Now that all unites (at t) of that collections might at another moment t* gather together and be in contact so that the whole collection becomes a UNIT at t*, by then it would be guaranteed an image {A,b}* , now the collection of all those images would be The set {{A,b}}. And so on...
This line of imaginary thought that highly simulate temporal topological properties of existing material in our world, does provide some motivation for having a size criterion upon sets. Clearly collections that are so big cannot have all its unites contacting to form a unit, and accordingly it might not have an image and cannot become an element of a class, thus would be a PROPER class. This is a reasonable motivation.
The above line of thought is what I called as: Imaging temporal Mereotopology.
Zuhair



