Set:
A set is a collection of objects in which order has no significance, and multiplicity is also ignored.

Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.

Let p be a point.

Let s be a segment.

The axiom of independency:
p and s cannot be defined by each other.

The axiom of minimal structure:
Any number which is not based on p, is at least based on s.

The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by p or s) of the entire collection.

(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.

Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html .

The axiom of completeness*:
A collection is complete if and only if its Urelement** cardinal and/or Quantitative** cardinal can be found, including the Quantitative** cardinal of its scale levels.

*Let us clarify the 'finite' concept in my framework:

In my system I have 4 building-blocks, which are: {}, ., ._., __ and the hybrid .__

The Quantitative** cardinal of {} is 0.

The Quantitative** cardinal of . is one of many.

The Quantitative** cardinal of ._. is one of many.

The Quantitative** cardinal of __ is The one.

The Quantitative** cardinal of .__ is one of two.

The bounds of lowest and highest existence (the ends) of these building-blocks ({} is excluded) are determined by their Urelement** cardinals, for example:

The lowest and highest bounds of . are Urelement** cardinals 0 and 0.

The lowest and highest bounds of ._. are Urelement** cardinals 0 and x>0.

The lowest and highest bounds of {} has no Urelement** cardinals.

The lowest and highest bounds of __ are Urelement** cardinals The 1 and The 1.

The lowest and highest bounds of .__ are Urelement** cardinals 0 and The 1.

Tautology:
The identity of a thing to itself.

Some details

‘=’ notation is used for both tautology (4=4) and quantity equality (4=2+2), which means that this system has to distinguish between elements not only by their quantity and/or order, but also by their structural properties, for example:

0 = .

1 = 0_1

2 = 0__2

3 = 0___3

4 = 0____4

are all tautologies, where 4 = 3+1 (for example) is not a tautology but quantity equality between the two different structures {0____4} and {0___3, 0_1}.

Now we can clearly see that in this framework we have two kinds of cardinals:

a)** Urelement cardinal: which is based on the lowest and highest bounds of bulding-blocks . , ._. , __

An example:
The lowest and highest bounds of {0____4} are the Urelement cardinals 0 and 4.

The Quantitative cardinal of {0____4} is the quantity 1.

b)** Quantitative cardinal: which is based on the quantity of . or ._. or . AND ._. that can be found in some set.

An example:
The Quantitative cardinal of {0___3, 0_1} is the quantity 2 .

Also, according to The axiom of duality, any arbitrary segment can be considered as {0_ ... _1}, and in this case the rest segments have their unique notations according to this segment.

The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.

The Axiom of the unreachable strong limit:
No input can be found by __ which stands for Fullness.

The Axiom of potentiality:
p is a potential Emptiness {}, where s is a potential Fullness __

The Axiom of phase transition:
a) There is no Urelement between {} and .
b) There is no Urelement between . and ._.

The axiom of complementarity:
Each number's structure which is not . or ._. is at least . AND ._.

The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by p or s) of the entire collection.

(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.

The Axiom of the unreachable strong limit:
No input can be found by __ which stands for Fullness.

The axiom of complementarity:
Each number's structure which is not . or ._. is at least . AND ._.

The Axiom of potentiality:
p is a potential Emptiness {}, where s is a potential Fullness __

The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.

The Axiom of the unreachable strong limit:
No input can be found by __ which stands for Fullness.

(a+b^n)/n = x

(http://www.godlessgeeks.com/LINKS/GodProof.htm)

A) An infinitely long entity is a non-composed non-empty scope which is not closed on itself and has no more than one reachable edge.

An example: __ , .__ , __.