Re: { }


David R Tribble wrote:
> Zuhair wrote:
> >> Also I find it very difficult to see why { } is not equal to nothingness.
> >
>
> leo1476 wrote:
> >> Do you mean "...equal to nothingness"? Well the empty set is a subset
> >> of every set because it logically follows from this argument:
> >
>
> 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.
>
> You're confusing concepts. { } means a set with nothing in it, but
> the set itself is not nothing.
>
> Consider a universe consisting of only fruits and boxes.
> Fruits can be placed into boxes, and boxes can also be placed into
> boxes (but obviously, nothing in that universe can be placed into
> fruits). We also assume that there are different kinds of fruits,
> and that a box containing fruits can only contain one kind of
> each fruit.
>
> Now if I have a box with three fruits (each one a different kind),
> this is the same as a set with three members, {apple,banana,cherry}.
> If I have a box with no fruits, this is the same as the empty set, {}.
> It's still a box (still a set), but it has no fruits (no members).
>
> In our universe, the concept of "nothing" is the same as saying
> no box or fruit (no set or member) at all. Obviously, "nothing"
> is not the same as an empty box (or empty set), because even
> an empty box (empty set) is something. "Nothing" is no "thing"
> at all.

Well in that case one should speak about Container theory or box
theory"lol".

The concept of "set" is in no way similar to the concept of
"container" you've
already demonstrated by your box example. The reason for your analogy
being wrong simply lies in sets being defined only by their members.
when I say the set of a and b
writtin as { a,b } it doesn't mean something which is containing a and
b in a sense that
it can contain members other than a and b and still remain the same
set.

The set of a and b is a and b together treated as a single unit.

If your intuitive container analogy was true, then the first axiom of
axiomatic set theory
_ two sets are identical only and only if they have the same members,
should be modified
towards_ two sets are identical only and only if they have the same
members and have
the same container.

So for example we should say that set X is defined as the container Y
which has
members a and b inside it.

in symboles { a,b } : { } = Y.

And if a set Z has members p and q inside it and a container W . then
Z=X if and only if
p=a and q=b and W=Y.

You might say that { } is the same for all sets . And so you should
believe in a sort
of a universal container symbolized by the two curely brackets. And
perhaps this
container is the transparent band created by our imagination around
entities, and it
seems to be the same band always. However what is the function of that
band.

In set theory that band should function as a collecter that tie
entities in one bundle.
this manage to make a different state when it collects multiple
entities together, but
what would it acheive if it ties one element? Moreover what such a band
acheive if
it try to contain no entity? In the later two cases the result with
that transparent band
or without it is the same, ie a single entity and a nothingness. So
your container
theory is not sufficient to explain set theory in a good intuitive way.

The problem is that traditional language about set theory doesn't
mention any defintion for a container at all.

The container can be regarded as the propositional function the
fulfillment of which cause
the members to be enrolled into a heap.

However this is not right because the same set can have multiple
propositional functions.

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" .

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.

Neither "no entities" nor "single entity" can form sets.

Also in a similar manner the concept of "All sets" is also not amenable
to set-hood.

So we cannot say the set of all sets, because simply such a thing
whould be itself always.

so if A= All sets then { A } = A , therefore A is not a set.

Zuhair

.

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