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 that determines

allowance of entry of objects into it; so a container that doesn't allow any

object to enter into it is a selective container and this is the empty set, on

the other hand a container that allows every object to enter into it is also

a selective container and that is the universal set, between those are containers

that allow some objects to enter into them and doesn't allow others from

entering into them, those are said to have a preferential selection rule.

By contrast if we imagine a container the doesn't have such a rule, for

example a container that may sometimes allow an object to enter into it

while other times it may not, those kinds of containers are examples of

containers that are not sets.

Allowance of entrance shouldn't be confused with ability of entrance;

rather allowance of entrance is to be understood as a kind of invitation for

entry, so whether the allowed (invited) object can enter the container or not

this doesn't affect the status of allowance of it! so a member (an element)

of a set is to be imagined as an object allowed by that fictitious container

to enter it even if it cannot actually enter that container so for example

we can understand that x is a member of x to mean that the fictitious

container x allows (gives permission to or invites) x to enter it despite

whether x can actually enter itself or not. This is generally similar to the

situation when one is invited to a party, so even if he actually cannot attend

the party still he is an invitee. 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 despite the actual ability of entrance of those.

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. Any set {x|phi} is said to be a *definable* set

because the selection rule of it can be defined as fulfillment of property

phi; "phi" here is named as a "predicate", so {x|phi} is a set defined after

the predicate phi, or sometimes it is called a *constructible* set, i.e. a set

that can be constructed after predicate phi.

However it is not always the case that the selection rule of a container can

be described in the above way, so containers that have a selection rule

that is not describable after a predicate are called *indefinable sets* or

*non constructible sets*, an example of that is a *Choice set*, for example

suppose that we have sticks belonging to Mr. Simpson and sticks belonging

to Mr. John, now suppose each has three sticks, now suppose Mr. Edwards

is to choose one stick from Mr. John's sticks and one stick from Mr. Simpson's

sticks, now suppose all those sticks look exactly alike, i.e. there is no

distinguishing feature that allows us to discriminate one stick from the

other in both groups, now it is only Mr. Edwards's choice that will

determine which stick is to be taken from each group and then put them

into a selective container, of course the selection rule is that of

Mr. Edwards's choice but this rule itself is not describable since there is

no distinguishing feature between those sticks, so the container into

which those chosen sticks are allowed to enter is a selective container

but the selection rule is not describable, we call that container the

Choice set of Mr. Edwards. On the other hand suppose that the condition

was different and Mr. Simpson had a red, blue and a green stick, also

Mr. John had a red, blue and a green stick, now this situation differs from

the above in that we do have a distinguishing feature that discriminate

between sticks in each group, suppose Mr. Edwards chose the red stick

from each group and put those into a selective container, now this

container is definable, it is {x| x is the red stick of Mr. Simpson's or x is

the red stick of Mr. John's}, so the selection rule is definable here, but in

the first example it was not, you see we cannot say for example that in

the first case we have a definable set defined as {x| x is a stick from

Mr. Simpson's or x is a stick from Mr. John's} because this will be the set

of all sticks (i.e. all six sticks), while Mr. Edwards only chose two sticks,

so it is not the same set. Although a choice set is a container whose

selection rule is not describable but that doesn't infer that it is not a selective

container, it is still a selective container, a selection has been made!!!

And membership of that set was dictated by that selection! But that selection is

not describable, so interpreting a choice set as a selective container is not out

of context. In a similar manner we can have other sets that do not have a

describable selection rule.

But which predicates can define sets? One would naively expect that any

property or any condition can serve as a predicate for defining sets after,

but this turned out to be fallacious, some predicates cannot define sets,

i.e., we can have a predicate phi such that we cannot construct any set

{x|phi}, the most famous of those predicates is the predicate: (x is not a

member of x), this predicate is the subject of Russell's paradox that would

be explained below. Now seeing that it is not the case that any predicate

can define a set, then we need to stipulate a collection of rules that determine

which predicates can define sets, and this collection of rules is what SET

THEORY is supposed to secure, this is done in an axiomatic manner, so

axioms of set theory lay out further characteristics of those fictitious

containers and set rules of defining them from predicates, and also

determine whether non definable sets are allowed to exist or not, etc…

So in nutshell: Sets are fictitious containers that possess rules of allowance

of entry of objects into them, those rules are called selection rules, those

might be total rejection or total acceptance rules or preferential ones,

may be definable rules or might not be definable; it is not the case that

every possible selection rule can be possessed by some container, that's

why we need a Set Theory so that we can rigorously determine which

selection rules containers can possess.

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 an aggregate *is* all of its 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. If a container

x allows the containers y1, y2 to enter then this doesn't entail that all of

what y1 and y2 allows to enter into would be allowed to enter into x; on

the other hand with the case of aggregates this not so, if you gather two

aggregates together then all elements of either aggregate would be elements

of the resulting aggregate.

So the concepts of "Aggregate" and "Set" need 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 if there is at least one set

that is not in itself then there must be a certain whole of sets that are not

in themselves, 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

V does 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 container 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 to form a certain whole of objects.

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?

Aggregates are the subject of "Mereology"; while selective containers are

the subject of "Set Theory".

Also as a piece of terminology common to Set Theory, in order to further

differentiate between containers that are elements of containers, and

containers that are not elements of any container, the term *Class* is

added, the rationale beyond that is for classes to stand for selective

containers as presented above, and those classes that are elements

of classes are to be termed as *Sets* while those that are not would

be termed as *proper classes*, this is the customary terminology used

in Set Theory, however I would suggest a better terminology that is one

that only uses the terms of set and element, so an element is what is

selectively allowed to enter into a selective container, a set may be an

element of a set and so it will be termed as a "set element", 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".

However those fine terminologies are not really important fundamentally.

The real fundamental issue is that of Sets and Aggregates.

To reiterate the definition of sets and aggregates:

Sets are fictitious selective containers.

Aggregates are wholes of objects.

This trial is of course an informal way of trying to engage the set concept,

however it proves to be an easy one to handle, and it definitely approximates

understanding of that concept to a great extent.

Zuhair Al-Johar

16/12/2011

Below are expositions of some 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: si is connected to si+1, and for each appropriate index j: sj, sj+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 sn=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 Gø 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 Gø 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.

For a more extensive treatment click here

Acyclicity Analysis:

A new project with the aim to find the relationship between cyclicity of

formulas and the strength of comprehension axiom schemes using them.

Acyclic comprehension has the strength of stratified comprehension which

is indeed very weak, however an observation that I made shows that only

adding one special kind of a cycle to an otherwise acyclic graph will pump

up the strength of comprehension using formulas with those graphs to the

level of having NF as a sub-theory of, it proves both infinity and transitive

closure, and of course define pure sets (sets with all elements in their transitive

closures being sets, where "set" is defined after Marcel Crabbe' in acyclic manner)

thus enabling interpreting NF. I'm of the feeling that minor cyclical modifications

results in big jumps in consistency strength of theories, this calls for finding a

rigorous system that classify cyclic formulas, try to find a measure of cyclicity,

and then relate those to consistency strength of theories using them.

A posting to FOM addressing this is present here.

Also the graphs of infinity and transitive closures are present here

September 22, 2012

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.

A variant of this theory is where public membership is defined after super-transitive closure

instead of transitive closure, and this is the set of all subsets of elements of the transitive

closure of a set. For details: press here

Another possibly related theory is present here

Predicative Set Theory: A theory defined in L(ω1,ω) where L is first order logic with equality

and membership, that I think it to be equi-interpretable to a subset of second order arithmetic

stronger than PA. It is Categorical! something that its finitary counterparts are not. Of course

the main feature is that it forms infinite sets in a predicative manner. The expressive power of

this theory is of course much stronger than its finitary counterparts.  So it combines Predicativity,

Expressiveness and Categoricity and is defined in a complete (and implicationally complete)

logical language that admits provability. A FOM posting is present here

The exposition of this theory is present here

Multi-level Discrimination Theory:

This theory involves working with very weak kinds of part-hood

relation and after them are defined very weak kinds of equality.

Axiom scheme III builds aggregates from atoms at the respective level

of discrimination. So it is limited in the sense that it cannot build

aggregates of non atomic aggregates at the same level, that's why

axiom IV is stipulated! it allows those aggregates to become atoms at

higher levels (of indiscrimination actually) and thereby they can be

gathered to form aggregates of them, this hierarchy if not inconsistent

could provide the necessary milieu for second order arithmetic to be

implemented in, thereby reducing most of mathematics to very weak

part-hood relations. Press here

Aggregate-Container Theory: This is still investigational, deep at the background of sets and classes.

Press here.

Finite Axiomatization of NF in four types: see here

Short axiomatization of SF: see here