Friday, August 15, 2008

Orders of Randomness 2

Orders of Randomness 2
Andre Willers
15 Aug 2008

See : “Orders of Randomness”

I have been requested to expand a little on orders of Randomness and what it means .
Please note that human endeavours at this date use only randomness of the order of flipping a coin ( Beth(0) )

Aleph is the first letter of the Hebrew Alphabet . It was used by Cantor to denote
Classes of Infinity (ie Aleph(0) for Rational numbers , Aleph(1) for Irrational Numbers , etc

Beth is the second letter of the Hebrew Alfabet . It means “House”

I will first repeat the derivation of Orders of Randomness from : “Orders of Randomness” because it is so important .

Start Quote:
First , simple Randomness .
Flip of a coin .
Heads or Tails . 0 or 1
Flip an unbiased coin an infinite number of times ,write it down below each other and do it again .
All possible 0 and 1’s

An example : Beth(0)
Flips(1) 0,1,1,1,1,… etc
Flips(2) 0,1,1,1,0,… etc
Flips(infinity) 0,0,0,0,0,0,…etc

This describes all possible states in a delineated binary universe .
“delineated binary” means a two sided coin which cannot land on it’s side .

Now draw a diagonal line from the top left of Flips(1) to Flips(infinity) .
At every intersection of this diagonal line with a horizontal line , change the value .
The Diagonal Line of (0,1)’s is then not in the collection of all possible random
Horizontal coin-Flips(x) .

This means the Diagonal Line is of a stronger order of randomness .
This is also the standard proof of an Irrational Number .

This is the standard proof of aleph numbers .
Irrational numbers ,etc
Since any number can be written in binary (0,1) , we can infer that the order of randomness is the same as aleph numbers .

This means we can use number theory in Randomness systems .
Very important .

Google Cantor (or Kantor)

Define coin-flip Randomness as Beth(0) , analogous to Aleph(0)
Then we have at least Beth(1) , randomness an order stronger than flipping a coin .
Then we can theorize Beth(Omega) <->Aleph(Omega) .

End Quote

Cardinal Numbers .

The cardinal number is the index x of Aleph(x) .
Cantor proved that
Aleph(n+1) = 2 ^ Aleph( n )

Where n is the cardinal number of the infinity .

Tying them together :
He also proved that
P(A) = 2^ n
Where A is any set , P(A) is the PowerSet of A and n is the cardinal number of set A
Thus , Cardinal Number of P(A) =(n+1)

The PowerSet of A = the Set of all subsets of A .
This sounds fancy , but it is simply all the different ways you can combine the elements of set A . All the ways you can chop up A .
You can see it easily in a finite binomial expansion (1+1)^n = P(A) = 2^n

See : “Infinite Probes”
There we also chop and dice , using infinite series .

Can you see how it all ties together ?

Why 2 ?

This derives from the Delineation Axiom . Remember , we can only talk about something if it is distinct and identifiable from something else . This gives a minimum of 2 states : part or non-part .

That is why the Zeta-function below is described on a 2-dimensional plane , or pesky problems like Primes always boil down to 2 dimensions of some sort .

This is why the irrational numbers play such an important part in physics .
Z=a+ib describes a 2-dimensional plane useful for delineated systems without feedback systems

Its in the axiom of Delineation , dummy .

But we know that Russell proved that A+~A smaller than Universum .
The difference can be described as the Beth sequences . Since they are derivatives of summation-sequences(see below) , they define arrows usually seen as the time-arrows .

These need not to be described a-la-dunne’s serial time , as different Beth levels address the problem adequately without multiplying hypotheses .

Self-referencing systems and Beth sequences .

A Proper Self-referencing system is of one cardinal Beth number higher than the system it derives from .
Self-referencing systems (feedback systems) can always be described as sequences of Beth systems . Ie as Beth(x) <-> Beth(y) . The formal proof is a bit long for inclusion here .

The easiest way to see it is in Bayesian systems . If Beth(x) systems are included , Bayesian systems become orders of magnitude more effective .

Life , civilization and markets are such . See below .

Conservation Laws :
By definition , these can always be written in a form of
SomeExpression = 0

Random (Beth(0) Walk in Euclidean 2-dimensions

This is a powerful unifying principle derived from the Delineation Axiom .

In Random Walk the Distance from the Center is = d * (n)^0.5 . This is a property of Euclidean systems .
(Where d = step , n=number of random beth(0) steps)

Immediately we can say that the only hope of the Walker returning to the center after an infinity of Beth(0) steps is if d ~ 1/(n)^0.5 . This is the Riemann Hypothesis .

Now , see a Universum of 2-dimensional descriptors z=a+ib

Sum all of them . Add together all the possible things that can be thus described .

This can be done as follows :
From z=a+ib Raise both sides to the e
e^(z) = e^(a) . e^i(b)
Raise both sides to the ln(j) power where j is real integers.
j^(z) = j^(a) . e^(b/ln(j))

Now , sum them :
Zeta=Sum of j^(z) for j=1 to infinity

Now we extract all possible statements that embody some Conservation Law . Beth(1)

This means that Zeta is zero for the set of extracted statements if and only if (b/ln(j)) is of the order of Beth(0) and a=(-1/2)

Tensors .
The above is a definition of a tensor for a discontinous function .

Riemann’s Zeta function.
This can describe any delineated system .
If Zeta = 0 , conservation laws apply .

Zeta = Sigma(1/j )^z for j=1,2,3,…,infinity and z=a+ib , where z is complex and i =(-1)^0.5
The z bit is in two dimensions as discussed above .

This function has a deep underlying meaning for infinite systems .
If you unpack the Right-Hand side on a x-yi plane you get a graph that looks like a random walk .

If every point is visited that a random walk would visit over infinity (ie all) , without clumping , then Zeta can only be non-trivially zero if a=(-1/2) .

Why (x – yi) plane ? See “Why 2 “ above . The system is fractal . Two dimensions are necessary in any delineated system .

Remember , randomwalk distance from origin = step*sqrt(number of steps) .
So if the steps = 1/ ( sqrt(number of steps) ) , then the Origin might be reached if and only if a= -1/2
This is easily proven .

If a= - 1/2 , then b can be any real function . This would include Beth(0) and Beth(1) , but not higher orders of beth .

If a= -1/2 and b is an unreal number , then a cannot be equal to -1/2 anymore . Conservation cannot hold at any level .

Conservation Laws can only hold for Beth(0) and Beth(1) systems .

This is forced by the two dimensions of delineation .

Mathematically , this means that Beth(2+) systems of feedbacks can only be described in terms of attractors or/and fractal systems (ie not in isolation)

Physically , conservation of energy and momentum need not hold for Beth(2+) systems .

This has an interesting corollary in decryption (unpacking) . A Beth(2) mind unpacking Beth(0) or Beth(1) encryption is functionally equivalent to Non-Conservation of Energy .

Some other consequences :
If a< -½ , then Riemannian Orbitals are described . Beth(any)
Also described as nuclei , atoms .

If a> -½ , then a diffuse cloud is described . Beth(any)
Also described as magnetic effects .

What does this mean?
Present technology uses Beth(x) technology in a rather haphazard way .(Quantum physics) .

A better understanding will bring about a sudden change in capability .


No comments: