Re: set of a set etc.

On 19 Jul 2005 07:55:28 -0700, "Jasper" <vfiddlestix@xxxxxxx> wrote:

>
> What is the difference between x, {x}, {{x}}, {{{x}}} etc?
> Especially for x = null set?
>

Let's consider x = {} and {x} for example. Now there is no element in x,
but there is an element in {x}, namely x. Hence there is obviously a
difference between x and {x}. This difference makes the difference. :-)

In math we adopt a special axiom for that, called /axiom of
extensionality/:

A = B <-> Ax(x e A <-> x e B).

So if there is an element in set A which is not in set B (or vice
versa), then A isn't identical with B.

Since there is no element in x, but x is in {x}, we actually have an
element, namely x, which is in one set, but not in the other. Hence:
x =/= {x}.

There you have your difference.


F.


>
> Actually I was wondering about the conceptual changes between different
> levels of "setting".
>
Conceptual changes?

Well we might ask how these sets are /defined/. One possibility is to
stipulate:

{a} =df {x: x = a}

Then we have:

{x} = {y: y = x)

{{x}} = {y: y = {x}}

{{{x}}} = {y: y = {{x}}}

:

With x = {} this means:

{x} = {y: y = {}}

{{x}} = {y: y = {z: z = {})}

{{{x}}} = {y: y = {z: z = {w: w = {})}}

:

x doesn't have an element. {x} consists of an element which doesn't have
an element. {{x}} consists of an element which consists of an element
which doesn't have an element, etc.


Hope this helps.


.

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