**Understanding
Sets.**

**Sets** are fictitious selective
containers.

By fictitious it is meant something
that is grasped by human imagery, it doesn't entail the non existence of those
matters in a manner that is separate from human existence.

A selective container is a
container that possesses a rule according to which it allows (or invites)
objects to be inside it and do not allow others from entering it. A member of a
set is to be imagined as an object allowed by that fictitious container to enter
it even if it cannot enter that container so for example we can understand that
x is a member of x to mean that the fictitious container x allows x to enter it
despite whether x can actually enter itself or not. Also we can understand that
x is an element of y and y is an element of x to mean that x allows y to enter
it and also y allows x to enter it. In general a set {x|phi}
is to be imagined as a fictitious container that allows every phi object to
enter into it and that do not allow any non phi object to enter into it.

Axioms of set theory lay out
further characteristics of those fictitious containers and set rules of
defining them from predicates. The case of indefinable sets is to be understood
as fictitious containers that have a selection rule that we cannot describe, so
they are to be understood as selective containers but describing this selection
is beyond the descriptive tools of the theory in question. A clear example of
that is a Choice set.

Do these fictitious containers have
a kind of existence that is separate from human imagery?

Nobody knows!

On the other hand **Aggregates**
need to be distinguished from Sets; confusion between Aggregates and Sets is
common since the difference between these two concepts is subtle. An aggregate
of objects is the whole of those objects so an aggregate of Mr. and Mrs.
Williams is both of them seen as one object, the pleural "the" often
refers to aggregates of objects, like "the books", "the
chairs", etc..., those refer to aggregate of books, aggregate of chairs,
etc.., and indeed the word "set" used in common language refers
mostly to aggregates rather than to sets as present in set theory. One cannot
imagine an aggregate of no objects, or an aggregate of a single object that is
different from that object, because aggregates "are" the wholes of
their elements, so an aggregate of one bird is that bird itself, it is not
something different from it, all of that illustrates the difference between
aggregates and sets, with sets we can have an empty set, because it is not
difficult to imagine an empty container, or more precisely speaking a container
that do not allow any object to enter it, also a container allowing only one
object to enter it is not necessarily identical to that object (unless it
allows only itself to enter into it), so singleton sets are not necessarily
identical to their sole elements. An aggregate of more than one object is
always different from its elements, but a set may have itself among its members
whether it was a singleton set or not. An example is the set of all sets, or to
rephrase it here the container that allows every container to enter into it.

So the concepts of Aggregate and Set needs to be discriminated. An example of
the importance of such discrimination is Russell's paradox which is often
stated as: there cannot be a set of all sets that are not in themselves, many
people would still insist that since there are many sets that are not in
themselves so there must be a certain whole of them, and therefore they would
argue that Russell's paradox is erroneous intuitively or is some kind of
language problem etc.., all of this is based on confusing sets as aggregates.
Russell's paradox is solved fundamentally by distinguishing sets from
aggregates, it should be read in the following manner: There do not exist a
container V such that every container x that do not allow itself to enter into
itself then it would be allowed to enter into V and such that every container x
that allows itself to enter into itself then it is not allowed to enter V.
Obviously this container do not exist, but that doesn't entail the non
existence of an aggregate of all those containers that do not allow themselves
to enter into themselves. Certainly this aggregate exist but also obviously
this aggregate itself is not a container! (unless
there is only one object that do not allow itself to enter into itself), so as
one can easily see the paradox pose no real intuitive problem. And to clear out
any possible confusion one must also distinguish between membership of sets and
membership of aggregates, those are distinct concepts, while the former is
allowance of entry to a container the later is being aggregated with objects to
form a certain whole of them.

Quite different from the case with
sets, aggregates of matters that have independent existence of human's do exist
independently, while with sets it is not so clear whether this is the case or
not?

Also as a piece of terminology
common to set theory, the term "set" is thought of as a
"class" that is an element of a class, while a class that is not an
element of any class is termed as a proper class, I don't see any need for
that, it is better to stick to one terminology that of sets only, a better
terminology would be to use the terms set and element, so an **element** is
what is an element of a set, a set may be an element of a set and so it will be
termed as a "**set element**" or simply as an "**element**"
if all objects in the relevant theory are sets, while a set that is not an
element of any set is to be termed as a "**proper set**". An
Ur-element is an object that can be an element of a set yet itself is not a
set; a better term to describe this case is a "**proper element**".

To reiterate the definition of sets
and aggregates:

*Sets** are fictitious selective
containers.*

*Aggregates** are wholes of objects.*

Zuhair Al-Johar

16/12/2011

For older approaches regarding
reality of sets see:

*Nature of sets at:* *pdf*

*Nature of sets_2 at: pdf*

Reality of sets at: *txt*

**Simplified graphical representation
of Sets:** Press here

Below is the exposition of some set
theories that I defined.

**Acyclic Comprehension Theory: **

**Definition of Acyclic formulae:**
We say that a variable x
is connected to a variable y in the

formula
ø* *iff any of the following formulae
appear in ø: x ∈ y , y ∈ x , x=y , y=x.

We refer to a function s from {1,…,n} to variables in ø* *as a chain of length n
in ø* *iff for each

appropriate
index i: s_{i} is
connected to s_{i+1}, and for each appropriate index j: s_{j}, s_{j+2} are two

different
occurrences in ø. A chain from x to y is defined as a chain s of length
n>1 with

s_{1}=x and s_{n}=y.

A formula ø is said to be
acyclic iff for each variable x in ø, there is no chain from x to x.

**Graphical definition of acyclic
formulae:**

With any formula ø* *associate
a non directed graph G_{ø}* *whose vertices are the
variables

occurring
in ø* *and which contain an edge from x to y for each atomic
formula

x ∈
y , y ∈ x , x=y , y=x which occurs
as a subformula of ø.

ø* *is said to be
acyclic iff G_{ø} is acyclic.

**Acyclic Comprehension: **For
n=0,1,2,… ; if ø

and
membership, in which y is free, and in which x does not occur, then:

∀w1…wn.∃x.(∀y.
y ∈ x ⇔ ø)

The full theory with the proof that
it is equivalent to NF\NFU, is present here.

**Disguised Set Theory:** this theory has an axiom scheme
that appears inconsistent

at first glance, but it proves very
difficult to find an inconsistency if any exists, the

basic idea is to define a new membership
relation that we call the public membership ∈

(as
opposed to the privet membership which is the primitive membership ∊) and
then stipulate

{x|ø}
exists if ø is a formula that only use predicates of equality and public
membership.

Public membership is defined as
privet membership of a set that is not a privet member

of the transitive closure of that
element.

X ∈ Y ⇔ (X ∊ Y ∧ ¬ Y ∊
TC(X))

This theory does prove infinity,
however it is hard to work with, mistakes very easily occur.

For details: press here

Another possibly related theory is
present here

**Hierarchy**** Set Theory**

HST
is just an alternative presentation of standard set theory, it interprets ZF in
a nice manner,

of
course HST is at least equi-interpretable with Morse-Kelley set theory, it
shows the model of

ZF as it is built and therefore it is a more elegant
presentation than the customary way of

presenting ZF which is etiological and ad hoc looking.
This representation shows that ZF is not

arbitrary at all.

HST is a pure class theory, no Ur-elements are
encountered, just sets and proper classes.

EXPOSITION

HST: is the collection of all sentences entailed by
first order logic with equality and

membership from the list of axioms outlined below the
following definition.

Define (set): set(x) ⇔ ∃y. x ∈ y

Axioms:

I.
Extensionality: ∀x.∀y.
(∀z. z ∈ x ⇔ z ∈ y) ⇒ x=y

II.
Impredicative Class
Comprehension: if ø is a formula in which x is not free then

(∃x.∀y. y ∈ x ⇔
set(y)ø) is an axiom.

Define ({|}): x={y|ø} ⇔
(∀y. y ∈ x ⇔ set(y)ø)

III.
Ordinals: Every accessible Von
Neumann ordinal is a set.

Define (V_{i}):

V_{0} = 0

V_{i} = P(V_{i-1}) for every successor set ordinal i

V_{i}=U(V_{j})
j<i for every limit set ordinal i

IV.
Hierarchy: For every set ordinal
i. ∀x⊂V_{i} . set(x)

Where P,U,⊂ denote Power,
Union and the Subclass relation respectively,

all defined in the standard
manner.

/

It is easy to see that all axioms of ZF are interpreted
here.

Define V= U({V_{i}| i is accessible ordinal}),
then define a ZF_set as

an element of V. Define binary relations ∈*, =*
as ∈ and =

restricted to V. State all axioms of ZF in terms of ∈*
and =*

over ZF_sets; it is easy to see that all those statements
are

proved in HST.

Zuhair
27/10/2011

Another short axiomatization
that proves Con(ZF) is present here

**My
way of writing First Order Logic. **Press here

Zuhair Al-Johar