What are sets? This is an old
question the answer of which is often left to the imagination of the reader.
Set membership in particular is stipulated as a primitive concept in formal set
theories; so the reader is left to fit in a suitable personal explanation that
conforms to how this concept is manipulated in formal set theories. Here is an
informal answer to this question followed by a complete formal workup referred
to by the links below.

We often imagine sets as aggregates
of their elements an example of such an aggregate is a heap of books where each
book can be imagined as an element of that heap, the heap itself is a kind of
an entity that describes the totality of all the heaped books. This line of
imagination doesn't tell us the full story of sets as present in formal set
theories, for instance a heap composed of a single book is that book itself
while in set theory a set with one element (member) may be different from that
element, we cannot imagine a heap that has no elements while we do have an
empty set, if we bring another heap of books and put it on top of the one we
already have then all books in either heap will be also elements of the sum
heap, while with sets this is not the case, a set of two sets may not have
members of each set as members of it.

So sets cannot be explained naively
as aggregates of their own elements. However the concept of an aggregate
clearly plays some role in the explanation of what sets are. Sets or more
broadly speaking classes can be imagined as special kind of *aggregates of
names*. So a set of for example book_1 and book_2 is to be imagined as an
aggregate of a name of book_1 and a name of book_2. Of course no name can name
distinct objects; however an object may have different names. For an aggregate
of names to be a set no two distinct names in it can name the same object, also
for various technical reasons every name is better be considered as an
imaginary indivisible object (an atom), so a name is more accurately rephrased
as a "**label**" but for convenience I shall keep using the term
"name" . Also for technical reasons an atom that is not a name of an
object is to be considered also as a class, so in nutshell a class is:

*a* *differential aggregate of atomic
names or an atom that is not a name.*

Now to understand set (class)
membership we need to view a member of a set not as an object aggregated in the
set but as an object a name of which is aggregated in the set. So if book_1 is
a member of a set B, then this is to be understood as a name of book_1 being an
atom aggregated to form the aggregate B. So classes are differential aggregates
of names of their elements. In this way we can clearly see how a singleton set
may not be identical to its element, and how can we have an empty set, and how
members of members of sets are not necessarily members of those sets. Also we
can understand how a set can be an element of itself, this is easily explained
as a set aggregating an atom that names it, so let's say that the set B is the
aggregate of two specific atoms: atom_1 and atom_2, and let's say that atom_1
is one of the names of set B, then set B would be a member of itself. Actually
all kinds of circular membership can be explained here. With this approach we can
also understand the possibility of having two distinct sets having exactly the
same members. This approach also helps to explain the difference between a
class and a set as present in formal set theories, where sets are stipulated to
be classes that are elements of classes, simply to translate this difference
here we stipulate that a set is a labeled class, i.e. a class that has a name.
Also this approach helps to explain the difference between a class and a
Ur-element, the later would be an aggregate of atoms that do not meet the
qualification for being a class like for example an aggregate where some
distinct atoms may name the same object i.e. a non differential aggregate of
atoms, or another example is an aggregate of many atoms all of which do not name
any object, or a mixture of those two kinds etc…, using this approach we can
actually define many sorts of Ur-elements.

For a rigorous formal account of
the above see:

Reality of sets at: *txt*

For older approaches to solve this
question see:

*Nature of sets at:* *pdf*

*Nature of sets_2 at: pdf*

**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

∀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

**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

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

**Finiteness
Mimicking Theory: **this theory proves the consistency of ZFC, so it can deal
with inaccessibility, the axiomatic background

is
motivated by the concept of mimicking what's happening

at the
hereditarily finite realm. For details: press here.

Zuhair Al-Johar