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Background Theory
Posted:
Dec 7, 2012 9:46 AM
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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 Part-hood "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 Part-hood 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 part-hood
is defined as non self part-hood, 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 descriptive-wise. 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.
Ur-elements 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. Part-hood: [for all z. z PP x -> z P y] <-> x P y
ll. Anti-symmetry: 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=[y|phi] <-> [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 {x|phi} will stand for the
Set representing [x|phi], { } 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, trans-coding 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
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