Re: Is {{}} a valid construct?

Neo wrote:
> > > Can an empty set contain an empty set?
> >
> > If a set contains an element, bit it the empty set or not,
> > its not the empty set.
> >
> > So put your one empty bag of potatoes into your other
> > empty bag of potatoes. And wush your second bag is not
> > empty anymore. You see, it contains the first empty bag.
>
> Thanks, however I am still confused as Bod Badour in
> comp.database.theory has the following reply to your post:

Why are you confused? The two answers say the same thing. {} is the empty set. {{}} is not the empty set. It is a set containing one object, that object being the empty set. Let's write x for the empty set. Then

x = {},
{{}} = {x}.

The set {x} clearly isn't x, nor is it empty.

> Sigh, I know I waste my breath but here goes:
> The empty set is the canonical (and only) set of cardinality zero: {}
> The set containing only the empty set is the canonical (but not the
> only) set of cardinality one: {{}}
> The set containing both the empty set and a set containing the empty
> set
> is the canonical set of cardinality two: {{},{{}}}
> And so on.
> One can research this further by searching 'formalism' in mathematics
> or
> by searching 'foundations of mathematics'.
>
> Neo needs to pay particularly close attention that the set {{}} is
> not
> empty because it contains {}.
> Another way of writing {} is =E2=88=85 ie. =E2=88=85 =3D=3D=3D {}
> Perhaps it would clarify if I rewrote the above sets as: =E2=88=85, {=E2=88=
> =85}, {=E2=88=85,
> {=E2=88=85}}
>
> =E2=88=85 is the empty set
> {=E2=88=85} is not empty because it contains =E2=88=85
> {=E2=88=85,{=E2=88=85}} is not empty because it contains both =E2=88=85 and=
> {=E2=88=85}
> etc.

-- 
David Marcus