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
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
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