An open domain is a domain that includes at least one non-finite collection.

A closed domain is any collection of finitely many finite elements.

For example:

{{},{},{{},{},{}}} is a closed domain

{{{},{},{},…}+{},{},{},{{},{},{}}} is an open domain.

An open domain does not have an Identity map.

A closed domain has an Identity map.

We have to understand that these conclusions, which are based on a research of the simplest possible collections, are definitely deeper, simpler and more rigorous than the Cantorean approach about the non-finite.

If you disagree with me, then you have to show how a non-finite and non-nested collection includes within its domain all of {} elements, while at the same time its Successor (+{}) exists out of its domain, because it has to be clear that if a Successor does not exist out of the domain of this collection, then this collection is definitely a finite collection.

Some example:

Let us say that we have a non-finite collection of non finite collections, for example:

{
{{},{},{},…}+{}
{{},{},{},…}+{}
{{},{},{},…}+{}
…
}+{{},{},{},…}+{}

As we can see, in this case the Successor is a one non-finite collection, which is out of the domain of the non-finite collection of non-finite collections.

This state holds in any nested degree, and we can clearly see that we get a fractal-like structure of infinitely many nested levels, none of which can be completed.

Another example:

Let us use Cantor's second diagonal method in order to prove that the Identity map of a non-finite collection does not exist:

Let us say that we have a non-finite collection which is composed of unique non-finite collections (where each non-finite collection has a unique order of empty and non-empty sets) for example:

{

{{ },{ },{ },{ },{ },...}
{{#},{ },{ },{#},{ },...}
{{ },{#},{#},{ },{ },...}
{{#},{#},{ },{#},{#},...}
{{ },{ },{#},{ },{ },...}

...

}

We can define another unique non-finite collection which is actually the non-finite diagonal opposite collection
{{#},{#},{ },{ },{#},...} , that has to be added to our non-finite collection, then we can define another opposite unique non-finite diagonal collection that has to be added to the non-finite collection, etc., etc. ... ad infinitum.

It is clearly understood that the Identity-map of a non-finite collection, which is composed by unique non-finite collections, does not exist, and the notation below is the general representation of this proof.

Q.E.D

{
{{},{},{},…}+{}
{{},{},{},…}+{}
{{},{},{},…}+{}
…
}+{{},{},{},…}+{}

x'≠x
x'≥x

"We know that in order to define 0 and 1 we need to define {} as the successor of itself."