# The meaning of set?

*From*: "zuhair" <zaljohar@xxxxxxxxx>*Date*: 25 Oct 2006 12:45:20 -0700

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. .

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

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

.

**Follow-Ups**:**Re: The meaning of set?***From:*Douglas Eagleson

**Re: The meaning of set?***From:*Rupert

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