Re: { }


David R Tribble wrote:
> Zuhair wrote:
> >> To my primitive intuitions one cannot say that { } is a set.
> >> Because {} leterally means_ Nothingness regarded as one whole.
> >> Now what is the difference between nothingness regarded as one whole
> >> and nothingness? To me zero * 1 = zero.
> >
>
> David R Tribble wrote:
> >> You're confusing concepts. { } means a set with nothing in it, but
> >> the set itself is not nothing.
> >
>
> Zuhair wrote:
> > The problem is that traditional language about set theory doesn't
> > mention any defintion for a container at all. [...]
> >
> > A set is defined by its members only, and equivalence between sets
> > depends on their members only , there is no definition for a container
> > that contains them. [...]
> >
> > Now { } has no members , and since any set is defined by its members,
> > then It seems more consistent to say that { } means "no set" .
>
> It would be more consistent to say that { } means "no members".
> Then you have to decide whether "no members" is the same as
> "empty set" or as "no set".
>
>
> > Again I say : set is a list of distinct entities considered as One whole.
> >
> > So, When there are no entities, then there are no sets.
> > When there are no distinct entities then there are no sets.
> > Only many entities can form sets.
>
> You are defining "set" as "a collection of one or more members".
>
> So how do you give any meaning to the intersection of two sets
> having no members in common, e.g., what is the meaning of
> A = { 1, 2, 3 }
> B = { 4, 5, 6 }
> A intersect B = ?
>
> Also, the following sets are disallowed by your set theory, even
> though standard set theory treats them all as different proper sets:
> { }
> { {} }
> { {}, { {} } }
> { { {} } }
> etc.

This is the second reply of mine on David R Tribble:

I like the box example David made. But if we consider the curly
brackets as that box
then sure every set has the same curly brackets and therefore the same
container or box.

If I should accept this analogy , then it would be more easier to
define the container as an element belonging to the set , and define
any set as a collection of two type of elements
a container element symbolized by the curly brackets { } which is the
same in all sets
and a contained element(s) , which lies inside the brackets.

In that way the first axiom of ZFC would be any two sets are identical
if they have the same contained elements.

the axiom of the null set would be , there is a set with one element
that is the container element , and that set has no contained elements
and this set is called the null set.

Of coarse it is clear that a set can have the same container elements
inside it more than once but cannot have the same contained elements
inside it more than once.

So for example { { a,b } , { a } } is a set , although the brackets in
{a,b} is the same
as the brackes in {a} but these are allowed to be repeated within a
set, but the contained elements are not allowed that property for
example { a, a} = {a} . while { {a} } =/= {a}
becaue the container element can be repeated within the set. while a is
not allowed
by that version of the axiom of regularity to contain itself. so
a=/={a}

In reality all the axiom of ZFC can be rewrittin using that
terminology, which I see it as a clearer terminology than the present
one.



Zuhair

.

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