Let us take for example the non-finite collection of the Natural numbers.

The Successor of this collection is notated as +1, because the simplest structure of the Natural numbers is the non-composed and non-finite collection that is notated as {1,1,1,1,1,…}+1, where +1 (the Successor) is the permanent next element, the existence of which was proven by Cantor’s second Diagonal method.

If the Identity map of a non-finite collection does not exist, then its exact cardinality does not exist and the Natural numbers’ cardinality is |N|-Successor, because the Successor is permanently out of our desirable “complete” domain.

Let @ be |N|-Successor

If A = @ and B = @-2^@, then A > B by 2^@, where both A and B are collections of infinitely many elements.

Also 3^@ > 2^@ > @ > @-1 etc.

So as we can see, in my universe I have both non-finite collections and unique arithmetic between non-finite collections, which its result is always a non-finite collection.

My results are richer than the Cantorean transfinite universe, for example:

By Cantor aleph0 = aleph0+1 , by me @+1 > @ .

By Cantor aleph0<2^aleph0 , by me @<2^@ .

By Cantor aleph0-2^aleph0 is undefined, by me @-2^@ < @ .

By Cantor 3^aleph0 = 2^aleph0 > aleph0 and aleph0-1 is problematic.

By me 3^@ > 2^@ > @ > @-1 etc.

|{{1,1,…}+1, 1,1,1}| > |{{1,1,…}+1}| by |{1,1,1}|.

|{{1,1,…}+1,{1,1,…}+1}| = |{{1},{1}}|•@ > |{{1,1,…}+1}| by |{1}|•@ and
|{{{1,1,…}+1, 1,1,…}+1}| = |{{1},1}|•@ > |{{1,1,…}+1}| by |{1}|•@ but they have different internal structures
( {{1},{1}} and {{1},1} ).

For further information, please read http://www.geocities.com/complementarytheory/Success... .

We can ask:

What is the difference between {}, {}+{}, {{}} or {{},...}+{}?

Answer:

{} is the empty set.

{}+{} is the permanent existence of {} as an empty set.

{{}} is some particular example of a finite and non-empty set.

{{},...}+{} is the permanent existence of {{},...} as a non-empty and non-finite set.

"The Successor of this collection is notated as +1, because the simplest structure of the Natural numbers is the non-composed and non-finite collection that is notated as {1,1,1,1,1,…}+1, where +1 (the Successor) is the permanent next element, the existence of which was proven by Cantor’s second Diagonal method."

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."