Re: set of a set etc.

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

It's obvious anyone can see the syntactical difference. I assume you
are asking a philosophical question.

First, assuming x is a scalar...

x is not {x} because x is a scalar and {x} is a set. Notice that
elements of {x} are unimportant. Given a scalar and a set, are they the
same? The answer is trivally no.

Now {x} and {{x}} are both sets and when comparing them their elements
do matter, i.e. we have to look inside. Given two sets, are they the
same? This is not trivial, we need to access their elements to complete
the comparison.

Notice that {x} and {{x}} are both sets. x and {x} are not both sets.
Can you see the difference?

If I say "for any set S..." then my assumption is valid for {x}, {{x}},
{{{x}}}, etc. but not for x.

Second, you ask what about x = null set. The null set is a set, so
really let's ask what about x = any set, as the answer is the same.
Then you have a list of sets, treat as above.

What I'm saying is, philosophically, there is no more difference
between {x}, {{x}}, {{{x}}} as there is between {a}, {b}, and {c}, in
both cases you have 3 unequal sets. In the former, you have the
coincidence that each successor contains its predecessor, and perhaps
even the bigger coincidence that it contains only its predecessor, but
you still just have different sets.

Contrast that to x and {x} and x and {p,q,r}, I would say x is no more
like {x} than it is like {p,q,r}. But you may ask since x is in {x},
isn't x more like {x}? The answer is no, while x is in {x} is true, x
is not in {x} is also true, there is no intrinsic reason here why one
of those statements is more valid than the other.

I hope this answers your question.

.

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