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 s₁=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 is acyclic formula in first order logic with identity 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

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):

V₀ = 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