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

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

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):

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

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

My way of writing First Order Logic. Press here

Zuhair Al-Johar