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Important: This topic is based on proofs without words ( http://mathworld.wolfram.com/ProofwithoutWords.html ).

A one rotation of the Archimedean Spiral is exactly 1/3 of the circle’s area ( http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Pappus/Bookiv/Pappus.iv.21-25/Pappus.iv.21_25.html#Prop.%2022 ):

If this area is made of infinitely many triangles (as can be seen in the picture below) , it cannot reach 1/3 exactly as 0.33333... cannot reach 1/3:

In order to understand better why 0.33333… < 1/3 please define a 1-1 mapping between each blue level of the multi-scaled Koch’s fractal that is found below, and each member of the infinitely long addition 0.3 + 0.03 + 0.003 + 0.0003 + … that is equivalent to 0.3333…

( http://members.cox.net/fractalenc/fr6g6s.577m2.html )

In any arbitrary level that we choose, the outer boundary of this multi-fractal has sharp edges.

0.333… = 1/3 only if the outer boundary has no sharp edges.

Since this is not the case, then 0.333… < 1/3.

Actually, we can generalize this conclusion to any 0.xxx… form and in this case 0.999… < 1 where 0.999… is a single path along a fractal that exists upon infinitely many different scales, where 1 is a smooth and non-composed element.

Now we can understand that a one rotation of the Archimedean Spiral is exactly 1/3 of the circle’s area only if we are no longer in a model of infinitely many elements, but in a model that is based on smooth and non-composed elements (and in this case the elements are a one rotation of a smooth and non-composed Archimedean Spiral and a one smooth and non-composed circle).

A model of infinitely many elements and a model of a non-composed element have a XOR connective between them.

Therefore the Cantorean aleph0 cannot be considered as the cardinal of N , because N is a collection of infinitely many elements that cannot be completed exactly as 0.9999... < 1.

In other words, by defining the Cantorean aleph0 as an exact cardinal of infinitely many elements, we are no longer in any relation with N, because N is based on a model of infinitely many elements and the Cantorean aleph0 cannot be but a non-composed and infinitely long element, which is too strong to be used as an input by any mathematical tool, and therefore it cannot be manipulated by the language of Mathematics.

Some words about Riemann's Ball:

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

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 ∞.

More information of this subject can be found in:

http://www.geocities.com/complementarytheory/ed.pdf

http://www.geocities.com/complementarytheory/Successor.pdf

I am a Monadist.

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).

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.

The foundations of Monadic Mathematics:

A scope is a marked zone where an abstract/non-abstract discussable entity can be examined.

An atom is a non-composed scope.

Examples: {} (= an empty scope), . (= a point), ._. (=a segment),

__ or .__ or __. (= an infinitely long entity).

An empty scope is a marked zone without any content.

An example: {}

A point is a non-composed and non-empty scope that has no directions where a direction is < , > or < > .

An example: .

A segment is a non-composed and non-empty scope that has directions which are closed upon themselves, or has at least two reachable edges.

An example: O , .__.

Each segment can have a unique name, which is based on its ratio to some arbitrary segment, which its name is 0_1.

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: __ , .__ , __.

Non-atom (or notom) is a scope that includes at leat one scope as its content.

An example: {{}}, {__}, { {},{{{}},{},{}},...}, {{{}} , . , ._. , ...} etc.

A sub-scope is a scope that exists within another scope.

An Open notom (or Onotom) is a collection of sub-scopes that has no first sub-scope and not a last sub-scope, or a one and only one infinitely long entity with no edges.

An example: {... ,{},{},{}, ...}, {__}, {... ,{{}},{},{}, ...} etc.

A Half-Closed notom (or Hnotom) is a scope that includes a first sub-scope but not a last sub-scope, or a last sub-scope and not a first sub-scope.

Also a Hnotom can be based on a one infinitely long entity that has at least one reachable edge.

An example: {{},{},{},...}, {.__}, {__.} etc.

A Closed notom (or Cnotom) is a scope that includes a first sub-scope and a last sub-scope, and it does not include Hnotom or Onotom.

An example: {{},{},{}}, {{}}, {{},{{},{{}}},._.} etc.

A Nested-Level is a common environment for a finite or non-finite collection of sub-scopes.

If a notom includes identical sub-scopes ( __ , .__ or __. are excluded), then it is called a First-Order Collection (or FOC).

An example:

{{},{},{},...}, {._. , ._. , ._. , ...}, {... ,{},{}, ...}, {... , ._. , ._. , ...}

{{},{}}, {{{}},{{}},{{}}}, {{.},{.},{.},...}, {{._.},{._.}} etc.

The name of an atom or a notom within some FOC is determined by its internal property and/or its place in the collection. From this definition it is understood that each atom or notom within a FOC, has more than one name.

Non-FOC (or NFOC) is a nested-level that does not include identical sub-scopes.

An example:

{{} , . , {} , ...}, {{._.} , ._. , ._. , ...}, {... ,{.},{}, ...}, {... , ._. ,{._.} , ...}

{{},{.}}, {{{}} ,{} ,{{}}}, {{},{.},{.},...}, {{},{._.}} etc.

Any atom ( __ is excluded) or notom has a unique name only if it can be distinguished from the other atoms or notoms that share with it the same nested level.

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

Let uncertainty be: more than a one unique name is related to an entity.

An edge and a point:

A point is a non-composed and non-empty scope that has no directions where a direction is < , > or < > .

An example: .

An edge is an inseparable part of an atom that has a direction.

An example: ._. , .__ , __.

A more developed version of this framework (but with different names) can be found in:

http://www.geocities.com/complementarytheory/My-first-axioms.pdf

Quote: |

you seem to reject the notion of limits. |

What is considered as a limit of some sequence that can be found upon infinitely many ordered scales, cannot be the limit of this sequence.

The reason is very simple, because if we examine the absolute value of the gap (the segment length) between any member in the sequence and the element that is considered as the limit of the sequence, we get the ratio 0_x/0 , where 0_x is the gap > 0 (which is a segment) and 0 is the gap between the limit to itself (which is a point).

From a point of view of a point (which is the hypothetic limit) each segment has the same length, and therefore nothing is converged to the point from the point's point of view, and the point cannot be considered as the limit of any segment.

The ratio 0_x/0 clearly gives us the notion that a point is not a limit of a segment (where a segment in this case is any gap > 0).

Instead of the limit concept, we can take any arbitrary segment and check the gaps (segments) relations of the sequence members, according to it (for example 0_x/0_s where 0_s is the arbitrary segment and 0_x is any member of the examined sequence).

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