Notice: This is a copy of my website at zaljohar.tripod.com
titled "Understanding Sets", it has been damaged their, that's
why a copy of it has been made here. This page copies
the home page of the site, links in it may not be working, so
please download them from the file attached to this page.
Zuhair Al-Johar
June 18 2016
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
addendum (27/1/2018): there is no need really to involve the "invitation" concept here just to explain self-membership, we can have the simple containment property, and define membership as being individual container contained in an individual container of the container. Though it adds one step deep to the definition of membership in terms of containment yet it solves the problem of having self membered sets.
I'd say that the "container" paraphrase of "set" is a very able one, it is a strong visualization tool to set processing, and so it is very helpful, but I think it is not exactly the essence of sets.
Classes can be formalized as Mereological aggregates of containers, and membership in classes as being a container that is a part of a class. Here this provides a distinctive visualization into classes and their membership compared to sets and their containment. I do really think that if we formally lay down an axiomatic theory about containers and their aggregates, then we can easily get to interpret full set theory, for example it seems somewhat intuitive here to have an infinite container, since all what we need is that at some moment of time there is a container in it, so we don't really need a very big container, since the size of containers is not a matter here, they need not have a fixed size all the time. A simple replacement axiom replacing each containee in a container by subcontainers of a common container would be enough [coupled with infinity] to interpret full ZFC.
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
What sets are about & About what sets are?: see here
MereoLogicism an explication of extensions and sets and membership: see here
Older Endeavors: Various redefinitions and reformulations
Of known theories and concepts. Press here