“Please show

which is not a member ofnN. If you can show it, then and only thenNis incomplete.”

The existence of * n* in

**N**is based on the axioms that define it, for example:

ZF axiom of infinity: If ** n** in

**N**, then

**+1 in**

*n***N**.

The ZF axiom of infinity actually states that **N**‘s collection of distinct members is incomplete inherently, since for any given ** n** in

**N**there is an additional (

**+1) member, and as a result no**

*n***in**

*n***N**is its final member. Order is not important here but Size is important.

If Size is the common property of **N** and **N** includes distinct members, then since **N** itself (The Size as a container) is beyond counting, no non-finite collection within it includes it, and as a result no collection is Size itself.

No Size depends on its content because it can be empty or non-empty and in both cases it is beyond its content or in other words, Size is a non-local ur-element that does not depend on any collection of finite or non-finite local members that are related to it in order to have a cardinal beyond 1. Since Size itself is beyond counting it is not equivalent to any cardinal that can be found as a result of the whole(Size)\parts(members) relation.

|**N**| != **N**(The Size itself) exactly as |{}| != {}(The Size itself) and in both cases no cardinal has the power of the container itself, and as a result, no cardinal of any collection (of distinct members or non-distinct members) is complete.

Since Cantor’s idea is based on a complete cardinal, then his idea does not hold if we understand that the whole (notated as {}) does not depend on the cardinality of its content.

The whole is empty **{}** or non-empty **{**{}**}** and in both cases it is beyond the cardinality of its content, for example:

**{****}** = Empty Size

**{**{}**}** = Non-empty Size

|**{****}**|=0 because **{****}** itself is beyond cardinality.

|**{**{}**}**|=1 because **{****}** itself is beyond cardinality.

|**{**a,c,b,d,…**}**| is incomplete because **{}** is beyond cardinality.

In other words, Size and cardinality are not the same, because Size is indepentent of cardinality and therefore cannot be measured by it.

This is a paradigm-shift in the foundations of Modern Mathematics and if you disagree with it, you have to show that Size = cardinality.

For example: Size = cardinality iff |{}|=0 and {} does not exist.

You can omit {} and use only ||.

In this case: Size = cardinality iff ||= 0 and || does not exist.

You can omit || and use the background itself.

In this case: Size = cardinality iff (the bachground itself) = 0 and (the bachground itself) does not exist.

So, please show that Size = cardinality.

(Since **{}** is not a member of itself (|**{**{}**}**|=1 because the outer **{}** is not a member of itself) we have a **natural solution **to Russell’s paradox and we do not need forced and un-natural thing like proper class, because the set of **all **sets does not exist, since the whole(or the Size) is independent and beyond the power(or magnitude) of its content).

[quote author=Jeroun]As long as *one* is the atom of itself, the cardinal can’t be more than one (The cardinal is the number of atoms of *one* & *one* is the atom of itself.) But if *one* wasn’t the atom of itself but something bigger, the cardinal in *one* could be higher.

It’s impossible to get a cardinal beyond one as long as *one* remains the atom of itself in my opinion.

In other words, in order to get cardinality beyond one, one has to be related to a thing that is beyond itself, which means that one is an open system.

By using the word “system” we mean that the one still exists when it goes beyond itself, and we get an organic structure, which is both a one thing that has many leafs.

In that case the one (or the system) is not “a one of many …” element, where a leaf is “a one of many …” element.

If we research the logical foundation that stands in the basis of a thing that is not “a one of many …” element, we can see that it is non-local by nature, and understood as NXOR or Not XOR connective.

In order to see it, let us research logically the concept of Membership, where its minimal condition is based on **in,out** relations:

Here is an example from set theory:

|{}| = 0 because only whet’s in {} is counted (only the “one of many ...” objects, or their absence (as can be seen in this particular example)) where the invariant whole (the container) is not counted.

It is o.k. that it is not counted, but not because it is ignored, but because it is the whole (the NXOR product, which is not “a one of many …” object)

As for the truth table of NXOR:

**in** **out**

0 0 → T (**in** , **out** are the same) = { }

0 1 → F

1 0 → F

1 1 → T (**in** , **out** are the same) = {* **}** *

As can be seen, we get T only if **in** , **out** are the same, and it does not matter if it is empty or full.

In this case we define a common logical basis for emptiness and fullness (they are two representations of the non-local element, or the whole, where the whole is not made of XOR products).

The logical basis of a part (a leaf) is based on XOR connective:

The truth table of locality is:

**in** **out**

0 0 → F

0 1 → T (**in** , **out** are not the same) = { }_

1 0 → T (**in** , **out** are not the same) = {_}

1 1 → F

The organic structure is based on NXOR\XOR logic.

Let system *Z* be the complementation between NXOR(non-locality) and XOR(locality).

The truth table of *Z* is:

**in** **out**

0 0 → T (**in** , **out** are the same) = { }

0 1 → T (**in** , **out** are not the same) = { }_

1 0 → T (**in** , **out** are not the same) = {_}

1 1 → T (**in** , **out** are the same) = {* **}** *

By system *Z* we may fulfill Hilbert’s organic paradigm of the mathematical language. Quoting Hilbert’s famous Paris 1900 lecture:

*“…The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.”*