Re: The meaning of set?


zuhair wrote:
Hi All,

The meaning of sets and set membership:

Sets can be intuitively thought of as conceptual containers having
clear cut inclusion and exclusion rules.

A container is what has the ability to contain some of what is other
than itself. The ability to contain is not determined by the existence
of contents within the container. The ability to contain is a de novo
property of the container itself.

Intuitively speaking one can say that what makes a container able to
contain others is the concept of "closure". For example this can be
shown well in geometric figures.
Geometric figures that can contain things within them should be closed
figures, open figures cannot contain things within them.

This closure is what makes a geometric figure able to contain other
geometric figures inside it.

However in the conceptual world, a concept is said to have closure if
it can hold a clear cut meaning, i.e. Dichotomous, in such a manner
that other concepts can be related to it in a clear cut manner.

More rigorously speaking closure is having a clear cut inclusion\
exclusion rules.

Inclusion rule to a container is the requirement for being contained
within it.
Exclusion rule of a container is the requirement for being not
contained within it.

Such containers if specified by their contents are called sets.

Set membership is the inclusion rule to the set.

A is a member of B mean that A fulfils the requirements of inclusion
into B.

So a member of a set is what fulfils the requirements of inclusion into
that set.

Simply speaking a member of a set is what fulfils its membership.


The Principle of conditional generality:

For every set there is a set which specifically has it as its only
member, and every
member of a set is a set of some other sets, unless sets are defined in
such a manner as to make this rule logically contradictive. .


From these simple intuitive ideas one can understand all axioms of set
theory.

1) Extentiality and the empty set are very clearly derived from the
meaning of sets.
2) Pairing: can be simply derived from the principle of conditional
generality as below.

If x is a set and y is a set, then the container which specifically
contains them as its sole members is by definition a set since it is a
conceptual container having closure
As defined by a clear cut set membership/exclusion rule.

3) Union, separation, replacement, infinity, choice, power set. All can
simply be derived from the meaning of sets mentioned above as far as
they involve no logical contradiction.

4) The axiom of regularity is inherit in the meaning of sets as
conceptual containers of others.

The set of all sets exist, but the set which contain it as a member
doesn't exist, nor does a proper superset of it exist, neither a power
set.

Though axioms in ZFC are rigorously followed in a blind manner, yet
giving flexibility to them so that they are only applicable within in
the confines of logic, is a better and a more reasonable approach.



Zuhair

Abstract container is alright as the correct style of symbol for set.

A cause to inclusion as the relation itself appears the difference of
the set you use and common set theory. You is better than common theory
except the fallacy of the red ball is always used to refute your theory
of set.

A set of All as abstracted red third order set is truely existenal as
long as the relation of red exists. SO when the red ball container is
made how does the inverted removal of balls occur?

A rule change is required.

So find the rule change to allow.

I am out of time. it is a complex subject you have. Some people spend a
lifetime of professional study and never understand the rather simple
inverted ball.

A red removal as the next, causes only red to be displayed out of the
container.

So true relative color has to be the cause of element in your theory.

And this then begins the walk to the Aristotle translation book.

Read it. And the set to be all as the abstract third order All element
a.

So I leave the comment with the abstract number dilemma displayed. All
as abstracted set inclusive must have its number relative to the
example set or the number can not relate all.

And the Paradox of Zeno is this challenge. So it is a big thing to
solve.

.

Relevant Pages

  • Re: The meaning of set?
    ... The meaning of sets and set membership: ... A container is what has the ability to contain some of what is other ... Inclusion rule to a container is the requirement for being contained ... So a member of a set is what fulfils the requirements of inclusion into ...
    (sci.math)
  • Re: The meaning of set?
    ... The meaning of sets and set membership: ... A container is what has the ability to contain some of what is other ... Inclusion rule to a container is the requirement for being contained ... So a member of a set is what fulfils the requirements of inclusion into ...
    (sci.math)
  • Re: The meaning of set?
    ... The meaning of sets and set membership: ... A container is what has the ability to contain some of what is other ... Inclusion rule to a container is the requirement for being contained ... So a member of a set is what fulfils the requirements of inclusion into ...
    (sci.math)
  • The meaning of set?
    ... The meaning of sets and set membership: ... A container is what has the ability to contain some of what is other ... Inclusion rule to a container is the requirement for being contained ... So a member of a set is what fulfils the requirements of inclusion into ...
    (sci.math)
  • Re: The meaning of set?
    ... The meaning of sets and set membership: ... A container is what has the ability to contain some of what is other ... Inclusion rule to a container is the requirement for being contained ... So a member of a set is what fulfils the requirements of inclusion into ...
    (sci.math)
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