S={x in A : x not in f(x)}

0 <--> {0,1} , 1 <--> {10,11,12} , 2 <--> {5,6} , 3 <--> {3,4,5} , 4 <--> {8,9}, …

In this example S ={1,2,4,…}.

S={x in A : x not in f(x)}

For some S, S={x in A : x not in f(x)}

For some S, S={x in A : x not in f(x)}

x not = x

Let us take another step and ask:

If the contradiction concept does not exist in Included-middle reasoning, then how can we check the consistency of its axioms?

After all, if there is no contradiction then there is no limit to anything and we cannot determine the consistency of anything in this framework.

My answer is this:

In Included-Middle reasoning any product is the result of constructive interactions between at least two opposites, so if something exists because of this interaction, it cannot be anything but a consistent product of this interaction, or in other words, inconsistent products simply do not exist in this framework, and all we have is consistent elements.

An axiomatic system which is based on an Included-middle reasoning, is based on the identity of a thing to itself, which is the new and simple meaning of the tautology concept in an Included-middle reasoning framework ('if, then' propositions are not needed here).

In short, all the "Energy" in an Included-middle reasoning goes to research what we can do with our existing elements, and we do not spend our "Energy" checking the existence of each element in our framework, because if it survives the interaction between two opposites, it cannot be but an existing (and consistent) thing in our mathematical framework.

Please look at this Included-middle reasoning axiomatic system: http://www.geocities.com/complementarytheory/My-firs... for further reference.

In Monadic Mathematics there are two separated models of the non-finite:

a) A model that is based on the term "infinitely many ...".

b) A model that is based on the term "infinitely long (non-composed) ...".

The Cantorean universe is based only on (a) model.

Because of this reason Cantor did not understand that when he use an AND connective between totality (the term 'all') and a collection of infinitely many ... , he immediately find himself in (b) model.

Please read very carefully my Riemann's Ball argument , in order to understand the phase transition between (a) model and (b) model (and vise versa).

Some words about Riemann's Ball:

By using Riemann's Ball we can clearly distinguish between potential infinity and actual infinity:

http://www.geocities.com/complementarytheory/RIMLIM....

As we can see from the above example, no infinitely many objects (where an object = an intersection in this model) can reach actual infinity.

In our example we represent only Z* numbers, but between any two of them we can find rational and irrational numbers.

Riemann's limits are 0 and ∞ (or -∞), and all our number systems are limited to potential infinities, existing in the open intervals (0,∞) or (-∞,0).

When we reach actual infinity, then we have no information for any method that defines infinity by infinitely many objects.

Also ∞ cannot be defined as a point at infinity in this model, because no intersection (therefore no point) can be found when we reach ∞.

If you understand Riemann's Ball argument then you can clearly see that Aleph0 cannot be but a (b) model.

Since there is a XOR connective between (a) model and (b) model, there is no relation between Aleph0, which is a (b) model, and set N, which is an (a) model.

"When we reach actual infinity, then we have no information for any method that defines infinity by infinitely many objects."

(F xor T) and (F xor T)

(F xor T)

(F xor T) and (F xor T)

(F xor T) xor (F xor T)

(F xor T) xor (F xor T)

A xor A

|___T XOR F
|
|___T XOR F
|

.____F
| |
| |___T

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).

Let redundancy be: more then one copy of the same entity can be found.

Let uncertainty be: more than one unique identity of the same entity can be found.

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)