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Background Theory

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Zuhair

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Dec 7, 2012, 9:54:05 AM12/7/12
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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 where 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

fom

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Dec 7, 2012, 11:21:48 AM12/7/12
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On 12/7/2012 8:46 AM, Zuhair wrote:

> 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!


I have no doubt that you are correct. In another post
in your thread I summarized the work of Lesniewski which
uses the part relation to characterize classes. His
method was specifically designed to circumvent the
grammatical form that leads to Russell's paradox.

Moreover, the historical record of the philosophy sides
with you. Aristotle, Leibniz, Kant, and Russell all
have had positions asserting how parts are prior to
individuals.

The issue becomes, however, whether the theory informs
differently. I turned to the investigation of parts
without any knowledge of mereology because I believe
that the ontological interpretation of the theory of
identity is inappropriate for foundational purposes.
Find any distinction in the literature between the
application of logic to a linguistic analysis and
the linguistic synthesis of a theory.

That is why my proper part relation has a the
character of a self-defining predicate. There must
be a first asserted truth, and, it must be an
assertion that constructs the language.

The second sentence has parallel syntax to the
first. It took me a long time to understand
why I should accept that situation. In Liebniz'
logical papers, one finds the remark that individuals
should be identified with a mark. The membership
relation is precisely that relation which marks an
individuated context. It does so in relation to
a plural context.

The priority of the part relation as it applies
to set theory is that the fundamental relation
between objects of a domain should reflect the
object type of the domain. To the extent that
sets are

"collections taken as an object"

the fundamental relation should relate collections
to collections. To the extent that a set is

"determined by its elements"

requires that the individuated context be situated
relative to the defining syntax of the fundamental
relation.

One of the primary focuses of my investigation
was to understand the principle of identity of
indiscernibles. If you read Leibniz, you will
find that what is taught about the identity of
indiscernibles is not entirely representative
of what Leibniz said.

He attributes his motivation to a position held
by Thomas Aquinas and explains that the typical
formulation is a consequence of the phrase


"an individual is the lowest species"


This is best interpreted mathematically by
Cantor's nested closed set theorem with vanishing
set diameter.


The following assumption:

Assumption of Aquinian individuation:
AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) ->
Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw)))))

can be used to define membership

AxAy(xey <-> (Az(ycz -> xez) /\
Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw))))))

relative to a prior proper part relation "c". In both
sentences, the complexity arises from capturing the sense
of a nested sequence of sets.

To be quite honest, here, I still need to consider
these expressions further. These universals are
difficult because both the assertion and its
negation must be considered.

So, let me suggest that while you continue
to refine your ideas, you also develop some of
the claims you have made concerning
the representations of arithmetic and such.

Do not take these remarks wrong. I see that you
are reading other authors and developing your
ideas. So, please continue. I would like to
see more.

Inform us.



























Zuhair

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Dec 8, 2012, 2:09:00 AM12/8/12
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Actually I'm contemplating another definition of set.

A set is an atom that uniquely represent a collection of atoms or
otherwise not
representing any object.


Set(x) <-> atom(x) & for all y.(x Rp y -> y is a collection of atoms)

in this way the empty set is an atom that is not a representative.

While Ur-elements can be thought of being representatives of gunk
(atomless) objects.

for example George Washington is not thought of as a collection of
descriptive atoms, so in the view of this theory it is atomless, so
the atom representing it might be thought of as an Ur-element and of
course it would be empty of members. However it would be different
from the empty set, since the later is about representing a collection
of atoms or about absence of representation altogether.

I think this is more faithful to the concept of empty set and Ur-
elements than the converse.

so for example the set {Miami} here Miami is the descriptive atom of
the city MIAMI, now the real city is not a collection of descriptive
atoms, it is a collection of physical atoms yes, but those are not
atoms functioning as descriptive units, so the real city MIAMI is seen
as atomless object, for which there is a descriptive atom, i.e. an
atom that represent it, this descriptive atom is named as "Miami"
now the set {Miami} is the descriptive atom of the collection [Miami]
in other words the set {Miami} is the atom that represent the atom
"Miami" that represent the real city MIAMI. of course Miami is empty
of members, but it is not yet the empty set since it does represent
something while the empty set does not, so it is an Ur-element.

I think this re-definition of set is more natural than the previous
one.

Zuhair

Zuhair

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Dec 8, 2012, 2:23:50 AM12/8/12
to
On Dec 7, 7:21 pm, fom <fomJ...@nyms.net> wrote:
> On 12/7/2012 8:46 AM, Zuhair wrote:
>
> > 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!
>
> I have no doubt that you are correct.  In another post
> in your thread I summarized the work of Lesniewski which
> uses the part relation to characterize classes.  His
> method was specifically designed to circumvent the
> grammatical form that leads to Russell's paradox.
>

Yes, this is clearly resolved here. A set would be an element of
itself iff
it represents a collection of atoms having it among them, this is not
that difficult
to ponder about since indeed a representative of a group can be among
that
group like in for example a father representing his family. Now the
collection of all
sets that do not represent collections of atoms of which they are a
part (i.e.
sets that are not elements of themselves) can be easily composed in
this
theory but also this theory easily prove that such a collection cannot
have
a representative atom. Set-hood is not about collections of atoms per
se,
it is about uniquely representing those by atoms. So as you see above
what seems to be a counter-intuitive result i.e. the non existence of
the set
of all sets that are not elements of themselves, is actually rendered
a quite
natural and intuitive result by the above line of thinking, that's why
I say that
the real benefit of background theory is that it makes one see the
whole picture
behind set theory, it reveals the whole background Ontology that is
usually hidden
from the customary presentation of standard set\class theories.
This background theory enable one to understand NAIVELY things that
otherwise
would be very difficult to grasp, like non well founded sets, non
definable sets,
non extensional objects, the interplay between classes and sets, the
paradoxes
etc... so to me it aids a lot in understanding those matters.

Zuhair

Zuhair

unread,
Dec 8, 2012, 3:57:09 AM12/8/12
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On Dec 7, 7:21 pm, fom <fomJ...@nyms.net> wrote:
I just wanted to note that with this approach a set is a singular
entity that
represent a plurality of singular entities, while with Lewis's
approach
a set is a plurality of singular entities that is represented by a
singular
entity. I see this approach reductive while Lewis's diffusive. However
formally speaking they almost mirror each other, but conceptual wise
I think the approach given here is more faithful to the general
context
in which sets are mentioned.

Zuhair

fom

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Dec 8, 2012, 6:13:29 AM12/8/12
to
On 12/8/2012 1:23 AM, Zuhair wrote:
> On Dec 7, 7:21 pm, fom <fomJ...@nyms.net> wrote:
>> On 12/7/2012 8:46 AM, Zuhair wrote:
>>
>>> 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!
>>
>> I have no doubt that you are correct. In another post
>> in your thread I summarized the work of Lesniewski which
>> uses the part relation to characterize classes. His
>> method was specifically designed to circumvent the
>> grammatical form that leads to Russell's paradox.
>>
>
> Yes, this is clearly resolved here. A set would be an element of
> itself iff
> it represents a collection of atoms having it among them, this is not
> that difficult
> to ponder about

You would be surprised. I find no problem with
using a part relation for first-order satisfaction.

The problem is simply in understanding that parts
are prior to individuals.

Nevertheless, the entire ontology of modern first-order
logic is based on interpreting the universal quantifier
over sets and understanding sets by the framework of
Russell's vicious circles and the hierarchical type
theory that arose from it.


<snip>
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