**Notion #1:**

If we use partitions in order to define Entropy, then a multiset (a repetition of the same identity) has an entropy that is equivalent to the number of the repetitions that exists within it.

Since a set has no repetitions, it has no entropy.

Let us examine the partitions that exist within any given * n* > 1

**{x} = Full entropy **

{x} = Intermediate entropy

**{x} = No entropy **

**2**

—-

**{1,1}**

**3**

—-

**{1,1,1}**

**{2,1} **

4

—-

**{1,1,1,1}**

{2,1,1}

**{2,2}**

**{3,1}**

**5**

—-

**{1,1,1,1,1}**

{2,1,1,1}

{2,2,1}

{3,1,1}

**{3,2}**

**{4,1}**

**6**

—

**{1,1,1,1,1,1}**

{2,1,1,1,1}

{2,2,1,1}

**{2,2,2}**

{3,1,1,1}

**{3,2,1}**

**{3,3}**

{4,1,1}

**{4,2}**

**{5,1}**

**7**

—-

**{1,1,1,1,1,1,1}**

{2,1,1,1,1,1}

{2,2,1,1,1}

{2,2,2,1}

{3,1,1,1,1}

{3,2,1,1}

{3,2,2}

{4,1,1,1}

**{4,2,1}**

{5,1,1}

**{5,2}**

**{6,1}**

...

As can be seen, Prime numbers have the least entropy, from this point of view.

Notion #2:

If we understand the **Sieve of Eratosthenes** ( http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes ) as a whole\part framework, than number 0 is the most dense part of it, and the set of primes is the least dense part of it.

In order to see it, let us represent the **Sieve of Eratosthenes** by non-finite frequencies notated by half circles, along a non-finite straight-line.

The first frequency is the non-finite collection of half circles that are representing the frequency level 1.

The next frequency is the non-finite collection of half circles that are representing the frequency level 2.

….

The next frequency is the non-finite collection of half circles that are representing the frequency level * n*.

Etc., … etc. …

Since the non-finite frequencies are synchronized with each other in Zero point, then 0 is the most dense part of the **Sieve of Eratosthenes**.

The least dense part of the **Sieve of Eratosthenes** is the set of prime numbers, because each prime number is a synchronization between no more than 3 frequencies, which are level 0, level 1 and the level of the prime itself.

Here is the diagram of the **Sieve of Eratosthenes**, represented as non-finite levels of synchronized half circles:

At the left side of this diagram we can see the Zero point, and the first 20 primes are mareked along the 0_level line.

———————————————————————————-

The non-local ur-element is the maximum entropy of itself (no differences can be found within it). Also a local ur-element is the maximum entropy of itself (no differences can be found within it).

Maximum entropy exists in both non-locality and locality, but they are opposite by their self nature, so if non-locality and locality are associated, then a non-entropic domain is created.

The history of such a domain is written by symmetry, where at the first stage symmetry is so strong that no product of this domain has a unique identity, and all we have is a superposition of identities.

Symmetry is collapsed because the opposite properties of non-locality and locality are expressed more and more until each local ur-element has a unique identity of its own.

This uniqueness, which is anti-entropic by nature, cannot exist without the association between the non-local and the local.

Opposite properties do not contradict each other, if they are based on NXOR connective.

A NXOR connective enables the existence of NXOR\XOR logic (non-locality and locality are associated, and associated realms have more than one entropy level).

A XOR connective does not enable the existence of NXOR\XOR logic (non-locality and locality are isolated, and isolated realms have maximum entropy).

Here is a part of my work called Eventors ( http://www.geocities.com/complementarytheory/Eventors.pdf pages 13-14):

*This paper expands the wave-particle duality to a whole universe, where a universe is a complementary phenomenon that exists between two opposite properties, which are integration (non-locality) and differentiation (locality). *

Integration is understood as gravity and differentiation is understood as expansion.

The most integrated state is understood as the 4D which is time(timing) or time-line.

The 3 other dimensions are the observed space, which is ordered relatively to the time-line that is considered as its attractor.

So the history of a universe is the story of space/time complementary associations along the time-line.

Without this time-line, no fundamental conditions can appear as natural laws of a universe.

With this model we can examine the idea of rich enough conditions in the space/time fabric, which could explain the origin and progression of life phenomena along the time-line.

The next part of this research is to develop a new fundamental mathematical language, using the insights coming from Quantum-Mechanics, where redundancy and uncertainty are first-order properties of its axiomatic system.

By doing this, we actually re-examine the whole scientific cosmological research in a new light, where the researcher himself is both observer_AND_participator.

From this point of view any result in any level (and not just in QM level) is influenced by the researcher, and the researcher has to include this influence as an inseparable part of his results.

*By using the word ‘result’ we mean that by this model, ethical results must also be considered as an organic part of the scientific research and development, where ‘development’ has two basics which are our technical skills and our ethical skills, which are combined into a one comprehensive scientific method, that can help us to survive the power of our developed technology along the time-line.*

I think that the organic approach (the associations between the non-local and the local) is the accurate way to understand the realm that we are an inseparable part of it.

—————————————————————————

Let us re-examine these cases:

Case 1: associated realms have more than one entropy level.

Case 2: isolated realms have maximum entropy.

In case 1 NXOR is associated with XOR and we get an open realm because both NXOR and XOR go beyond their self state of maximum (and opposite state of) entropy.

In case 2 there is no association between NXOR and XOR, and each opposite is closed upon its own maximum entropy, and nothing exists beyond these closed and isolated opposite maximum entropies.

In a complementary realm, each opposite is opened to an “off spring” product, which is beyond its own isolated state (an isolated realm has maximum entropy).

About dimensions:

If an organic realm is the result of the associations between the non-local and the local, than our measurement tools must express this association.

For example, let us take the place value method.

If we look at it from both parallel and serial points of view, we get a fractal-like structure, which is a mixed pattern of both parallel and serial parts upon finite/non-finite scales.

Let us examine this structure by using bases 2,3 and 4:

The traditional place value system is based only on the serial broken-symmetry building-block, which is used to define non-finite fractals upon non-finite scale levels, where the structure of each fractal is determine by the serial broken-symmetry building-block that is used.

Furthermore, the traditional method ignores the whole/part relations that exists in such fractals and uses single paths along them as measurements tools, for example:

Pi representation in base 10 is a single path along a base 10 fractal, and this single path is notated as 3.14159265358979323846264338327950288419716939937510 …

where each numeral represents a different scale level along this fractal.

The organic approach changes at least two things here:

1) The fractal-like structure is based on both parallel and serial building-blocks.

2) There can be simultaneously more than a one path , and as a result our measurement tool is not limited to a single path of numerals, but it can be a tree of several paths made of several building-blocks with different symmetrical states, which simultaneously determine the structure of what I call Organic fraction. Here is an example of an organic fraction that is based on different bulging-blocks taken from bases 2,3 and 4:

So as can be seen, the 4D model is just the standard approach to start with.

In order to deal with Organic fractions, a parallel/serial Turing-like model has to be formulated.

I am in a state of “Michael Faraday”-like* here that seeks for “James Clerk Maxwell”-like** in order to do that.

* http://en.wikipedia.org/wiki/Michael_Faraday

** http://en.wikipedia.org/wiki/James_Clerk_Maxwell

I think that since non-locality is involved here, then any formulation of Organic fractions must be incomplete and therefore open (this is a positive interpretation of Gödel’s work).

Please read this message to Prof. Mandelbrot http://www.geocities.com/complementarytheory/2Mandelbrot.pdf .

In my opinion, meaningful frameworks exist as long as there is a difference between X-model and X (which is also a positive interpretation of Gödel’s work).