Re: The meaning of set?


Rupert wrote:
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

Sorry, I'm not clear on what alternative foundation you're proposing.
Are you proposing a different axiomatic theory?

What I am proposing is a different approach to sets and analysis based
on them.

The ironic , congrete thinking like processing that is present in ZFC
and perhaps other set theories is yach, strangulating.

I want a set theory that differentiates between what I call Necessary
characteristics for Identity and un-necessary characterisitcs. If Moe
Blee comes every day wearing a black trouseurs, it doesn't follow that
if one day he wears a white one then he is not Moe Blee.

If water is evapourted, or freezed to make ice, it is still WATER.

But if I cleave one hydrogen itom from H20, then I will have )OH- and
that is not water.

Unless Sets and set membership is clearly defined, no theory can
determine what characteristics are essential to identify x as a set.

From my piont of view only two axioms of ZFC are necessary for the
identity of a set, and these are the axiom of extentiality and the
axiom of regularity. the others are not essential characteristics of
sets. The other axioms are to be treated with Flexibility.

Zuhair

.

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)