Emptiness {} and Fullness {__}, are the natural limits of any language, formal or informal.

Let Emptiness be represented by {}

Let a point be represented by {.}

Let a segment be represented by {._.}

Let Fullness be represented by {__}

By {} ← x(={.}) we mean that {.} is a potential {}.

By x(={._.}) → {__} we mean that {._.} is a potential {__}.

Monadic Mathematics is:

({},{__}):={x|{} ← x(={.}) AND x(={._.}) → {__}}

A set is only a framework to explore our ideas.

The concept of an oredered set does not depend on the quantity concept as shown here:

By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication
and addition are complementary opreations that can be ordered
by different internal symmetrical degrees where the quantity
remains unchanged, for example:

A Number is anything that exists between ({},{__})

Or in more formal definition:

({},{__}):={x|{} <-- x(={.}) AND x(={._.})--> {__}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.

This fading transition can be represented as:

(1*4)..............={1,1,1,1}.<-------------.Maximum symmetry-degree,
((1*2)+1*2)........={{1,1},1,1}..............Minimum information's
(((+1)+1)+1*2).....={{{1},1},1,1}............clarity-degree
((1*2)+(1*2))......={{1,1},{1,1}}............(no uniqueness)
(((+1)+1)+(1*2))...={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)..........={{1,1,1},1}
(((1*2)+1)+1)......={{{1,1},1},1}
((((+1)+1)+1)+1)...={{{{1},1},1},1}.<------ Minimum symmetry-degree,
..............................................Maximum information's
..............................................clarity-degree
..............................................(uniqueness)

First let us examine a well-known relation between mathematical objects and their representations.

=>> is ‘represented by’

|{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ;

|{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; …

Let us see how we use my method to construct a collection of R members.

R members are constructed like this:

1) First let us examine how we represent a number by my system:

=>> is ‘represented by’

a) |{}|=>>0

b) There is 1-1 and onto between ‘0’ and the left edge of {._.} and we get {‘0’_.}

c) |{{}}|=>>|{0}|=>>1

e) There is 1-1 and onto between ‘1’ and the right edge of {._.} and we get {‘0’_’1’}

In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.

2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’
and _’1’. Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.

3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:

a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.

b) We also have our initial domain _1, which stands at the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment, which is smaller or greater than the initial domain _1 , for example:

0 = .

1 = 0__1

2 = 0____2

3 = 0______3

.5 = 0_.5

0_________pi

The negative numbers are the left mirror image of the above numbers.

There is no division in my number system because both {.} and {._.} are indivisible by definition.

In short, any segment is an independent element that clearly can be shown in the above 2-D representation.

If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation,
for example:

0_.5__1_____2_____3__pi

Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.

oo=|{___}|

0=|{}|