Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Background Theory
Posted:
Dec 7, 2012 9:46 AM


The following theory is defined with the aim to provide a background Ontology in which set\class and their membership can be interpreted in a somewhat faithful manner.
Background theory is defined in first order language with equality. Added primitives are of Parthood "P" and Representation "Rp", both are binary relations both have many natural examples and can be grasped naively on informal level, and formalization to capture those informal concepts is easy and straightforward.
Natural examples of Parthood relation are seen in common day life like in saying that a brick is part of the wall, a window is part of the house, a finger is part of the hand, the heart is part of the body, the first name is part of the whole name etc... Proper parthood is defined as non self parthood, so we have x proper part of y iff x part of y & x=/=y. Now its pretty natural to assume that if an object has all its proper parts as parts of some object then it would itself be also a part of that object, the opposite direction is also very natural, this is only common sense, also it is natural to assume that no distinct objects can be parts of each other. Those principles are very natural indeed. An atom is an object that does not have proper parts. Atoms naturally exist in our descriptive discourse in common day life, a person is actually composed of many parts but we refer to her/him as one entity, i.e. we give it a singular impartation. This singular descriptive status is an atom descriptivewise. An overlap between objects is a part those objects share. A "collection of atoms" is an object where every part of it overlap with an atom. So every part of a collection of atoms is also a collection of atoms belonging to that object. It is also natural to assume that if x is a collection of atoms and all atoms of x are atoms of y then it follows that x is a part of y. Now the strong atomistic composition principle states that for any property phi that holds of some atom, then there exist a collection of all and only those atoms that satisfy phi. All of those principles are pretty much natural and they constitute the Mereological part of this theory.
Representation relation is also one that has many natural examples, a father representing his family, an attorney representing his client, a representative of a company, An ambassador representing his country, etc... Here we will deal with a special kind of representation relation that is called unique representation denoted by "Rp". Here an object can only have ONE representative, and each representative only represent one object. Also in order to encounter a description of the empty set we need to fix some atom to be representative of some unique atomless object, so we will axiomatize the existence of a unique atom that represent some atomless object. And Finally in order to be able to define relations we'll axiomatize that each binary collection of atoms have an atom that represent it, however these last two principles do really belong the set realm that this theory is aimed to provide a background for.
Now a set will be DEFINED as an atom that uniquely represents a collection of atoms or absence of atoms.
Define: Set(x) iff atom(x) & Exist y. (y is a collection of atoms OR y is atomless) & x Rp y.
Set membership will be DEFINED as being an atom of a collection of atoms represented by an atom.
Define: x member of y <> Exist z. z is a collection of atoms & y Rp z & atom(y) & x atom of z.
Urelements and proper classes can be defined in many ways in almost flawless manner.
FORMAL presentation of Background theory:
Background Theory is the collection of all sentences entailed (via rules of FOL(=,P,Rp)) by the following non logical axioms:
Define: x PP y <> x P y & ~ y P x
ID axioms + I. Parthood: [for all z. z PP x > z P y] <> x P y ll. Antisymmetry: x P y & y P x > x=y
Def.) atom(x) <> ~ Exist y. y PP x Def.) x atom of y <> atom(x) & x P y. Def.) c is a collection of atoms iff for all y. y P c > Exist z. z atom of y. Def.) g is atomless <> ~ Exist x. x atom of g
lll. Atomistic parts: [x is a collection of atoms & for all z. z atom of x > z P y] > x P y
lV. Representation: x Rp c & y Rp d > (x=y<>c=d)
V. Composition: if phi is a formula in which y is free but x not, then ([Exist y. atom(y) & phi] > [Exist x. x is a collection of atoms & (for all y. y atom of x <> atom(y) & phi)]) is an axiom.
Define: x=[yphi] <> [x is a collection of atoms & (for all y. y atom of x <> atom(y) & phi)]
For convenience writable finite collections of atoms shall be simply denoted by a string of those atoms embraced within solid brackets [], so [a] is the collection of atoms, that has one atom which is a, of course [a]=a; similarily [a,b] is the collection of atoms a and b.
Define: Set(x) <> atom(x) & Exist y. (y is a collection of atoms or y is atomless) & x Rp y
Define: x member of y <> Exist z. z is a collection of atoms & y Rp z & atom(y) & x atom of z
The curly brackets shall be used to denote SETs, so {a,b,c,...} stands for the Set representing [a,b,c,...], also {xphi} will stand for the Set representing [xphi], { } stands for the empty set.
/
That was the Background theory on top of which one can add set\class axioms and see all the background ontology behind sets and classes.
Now set axioms is to be added on top of this theory. So for example ZFC axioms relativised to Sets as defined here and with epsilon replace by the defined membership relation here all can be added to the above theory and this results in a theory that prove the consistency of ZFC. The real benefit is that one would see the whole background Ontology. Also it provides a nice philosophical interlude into what sets are? Sets are descriptive singular units (atoms) of pluralities, so the set of George Washington and Obama is the descriptive atom of the collection of the descriptive atom of George Washington and the descriptive atom of Obama.
So here sets are understood to rise within the discourse of describing pluralities in a singular modality. A Hierarchy of singular\plural interplay is what set theory achieves, thus providing a powerful logical background in which mathematics can be implemented.
Lewis chose another equivalent approach to the above, I myself have written many theories using Lewis's approach. His approach is to define sets and classes as pluralities of labels of pluralities, the labels are generally taken to be singular entities (atoms). This approach is weaker than the following as regards the context of singular description of pluralities, it is actually half way into that approach, it prefers to keep the plurality status of sets and the singular only play a weak intermediate role in defining a membership relation that is the basically between pluralities. This is so timid approach, and becomes very awkward if it tries to encounter the empty set, actually Lewis's approach better fits rejection of the empty set, otherwise the whole approach will turn to be very cumbersome, though possible of course but too undesirable.
Lewis's approach is to define set membership relation (accommodated to encounter the empty set) in the following manner:
x member of y <> y is a set & Exist z. z lable of x & atom(z) & z part of y.
set(x) <> x is a collection of atoms & 0 Part of x
where 0 is some fixed atom that is not a label.
This yields a lot of non set objects by the powerful composition principle. Also it looks awkward to fix every set to have the atom representing the empty set as part of it.
This approach becomes much simpler if one easily reject the empty set and simply define set membership as:
x member of y <> Exist z. z label of x & atom(z) & z part of y.
Anyhow as I said, this approach is too shy as far as the descriptive concept mentioned above is concerned.
There is a sense that the set concept is stronger than that, and that it is as I depicted here about atomic descriptions of pluralities, more than it being about some half way merely technical relation between pluralities to achieve a hierarchy of pluralities on the shoulders of singular intermediates that has no clear philosophical justification. With Lewis's approach one feels about walking in a jungle of pluralities linked to each other by singular links, so it augments plurality. While with the approach given here there is some feel of reductionism were pluralities are described by singular entities, so it lessens plurality, transcoding them into singular discourse which suits more the general context of speech about sets.
One might wonder if it is easier to see matters in the opposite way round, i.e. interpret the above theory in set theory? the answer is yes it can be done but it is not the easier direction, nor does it have the same natural flavor of the above, it is just a technical formal piece of work having no natural motivation. Thus I can say with confidence that the case is that Set Theory is conceptually reducible to Representation Mereology and not the converse!
Zuhair



