The Inside of Zero .

Andre Willers

7 Aug 2009

Synopsis :

A system of 13 Diophantine equations with 26 unknowns are the necessary sufficient to describe Arith I systems relative to Arith II system , with a Degree of Complexity = 10 .

These are used to describe a mathematical vacuum , with some physical consequences .

Discussion :

See Appendix A , B , Recursive Theory .

See previous posts , where Arith I and Arith II systems were discussed in detail .

The problem lies in discussing Non-Aristotelian systems using Aristotelian concepts of delineation (ie True , not-True ) .

Infinity .

The alert reader would have noticed that most of the problems come from processes continued indefinitely , which is taken as infinity . But is it ? Kantor already proved that varieties of infinity exists . It immediately follows that the software-computer we call mathematics and logic needs some revision .

The works of Russell , Godel , Matiyasevich et al pointed out some further contradictions in the Aristotelian model .

Can a theorem be true only for Aleph0 but not for Aleph1 ?

This is analogous to the problems with parallel lines continuing "infinitely" , that led to non-Euclidean geometries .

Recursive Genesis .

The standard axioms of arithmetic needs only a tweak on one axiom to generate the necessary revisions .

Generate new numbers by adding 1 to any number a .

Arith II

The Standard Set (call it Arith II) states that a+1<>a , where a is a previous number . The number line does not loop back on itself .

Arith I

The number line can loop back on itself . A circular number-line is formed . In a certain sense , we are discussing the topology of circular number loops in a Arith II space and their relationships .

The metric has not been defined . The question then becomes :

How many Arith I systems (= ArithI(m) ) plus one ArithII system (we only need one ArithII) are necessary sufficient to describe this particular Universum ?

Rotational Translations (spin) .

This is actually moving from one dimension to another , regardless of the frame of reference . Every ArithI system then actually needs a spin indicator : ie , which way it is curving in an (n-1) dimensional space .

I draw your attention to the curious fact that the angle in 2-dim is 2pi , while in 3 dim it is 4pi . More of this anon .

The Degree :

The maximum exponent in an equation if you change all the variables into one variable . This is important because it indicates the number of dimensions we have to use to describe the equation . Do not confuse it with the number of variables .

Minimum Necessary Sufficient .

The Ball-Breaker . The description defines reality .

This has been called many things :

Principle of Least effort , time , distance ,

Entropy .

Occam's Razor .

Collapse of the wave-function .

Economy of effort , etc .

The trade-off :

Matiyasevich et al has shown that there is a relationship between the Degree and the Number of variables necessary to describe an item in an Universum using a related number of equations .

Boundaries :

The following relationships has been proved :

Degree = 4 , variables 58

Degree = 10 , variables = 26 , equations =13

Degree = 10^45 , variables = 10

Is there a minimum number number of degrees ?

I doubt very much whether a Degree lower than 4 will be found . See Physical significance below .

See previous posts .

Physical Significance .

"Everything that can be , will be . But not all at once ." AW

The Degree can be described as the number of dimensions . You will notice the correlation with string theory .

Sadly , a Theory of Everything is impossible . But we can creep up on it .

Delicious !

Degree = 10 , variables = 26 , equations =13 , Spin =2

The numbers 26 =2x13 , and spin =2 should be knocking at the jaded doors of your mind .

Cards .

A pack of 13x4 = 52 cards forms a very good analogue of the Mathematical Process of a Universum .

You can work out for yourself why humans have a good use for a very good analogue of the universe .

And the Jokers ?

Remember , the Joker can take on any value . A good decription of a trans-luminal , low-probability event .

The most popular string theory uses 10 dimensions .

And the rest of the Tarot pack ?

Remember , we are talking about necessary sufficient without straining human capabilities too much .

Prime Numbers :

A prime number is simply an ArithI system (in ArithII measurements) that cannot be chopped up .

A mathematical atom , relative to ArithII . The number we need is related to the number of variables .

It is like zero

The Inside of Zero .

Degree = 10 , variables = 26 , equations =13 , Spin =2

If we plug in 26 prime numbers into the Diophantine polynomial generational equation in AppendixA below (and there are an infinity to choose from) , we get 26 ArithI systems , which have a mathematical vacuum inside them . No numbers .

A very interesting place . Note that the resultant is also a prime atom . It is recursive . Only the spin remains free .

Like the inside of a singularity .

Physically , this will have some very interesting effects .

There are no quantum fluctuations inside zero . The metric does not exist , even at Planck level .

Super-conductivity :

Purely an effect of the number of atoms crowded together .

It does not matter which atoms . They just have to be in certain configurations . Hence the present confusion in the field .

Disintegration of matter

(cold-fusion or cold-fission) .

But observational systems really like conservation laws . Energy release can then be only through particle or EM means .

If the geometries are chosen correctly , we can constrain the output mainly to electron/proton or electron/EM .

Direct electrical energy from matter . Very good power generation in our Universum .

Quantum Epigenetics .

The patterns on the surface of zero are constrained by trans-luminal effects inside zero . The outside patterns dictate the quantum-fluctations , as well as trans-luminal and super-luminal effects from all over .

The spin of Zero will thus drag creation of quantum fluctuations around it . This will affect things not only on a small scale , but on a large scale as well . The Drags do not balance out .(cf Relativistic rotation drag)

This can actually easily be calculated in the standard way by wave functions and General Relativity .

Rotating around a point

Note that there is a difference between spin and a particle rotating about center .

This can be constrained by using the fact that angular radians in 2 dimensions is 2pi and in 3 dimensions is 4pi .

Physically , in our descriptions , it means the particle does not really know whether it is orbiting in 2 dimensions or is spread over a surface in 3 dimensions (cf h/2pi)) , but we can constrain the geometries (and do in our quantum devices !)

God's sense of humour .

Degree = 10 , variables = 26 , equations =13 , Spin =2

Each degree (ie dimension) can take on +1, 0, -1 spins . Thus 10^3 number of states .

(We do not worry about minimum necessary sufficient spins , only state what is .)

This gives a polynomial of 27 integers of degree 10 with a value of 3 spins . See Appendix A below .

The Fine structure constant of our universe is

1/alpha = h/2pi * c / e^2

=137.035 999 070 (98)

where h is Planck's Constant , c is lightspeed in vacuo (see above) and e is electron charge , all in dimensionless electrostatic units . The value is dimensionless (ie the same for any definition of units)

It shows the relationship between h (Plancks constant , which includes the definition of mass) , spin (the pi , but there has to be compensation for dimensional drifting between dim2 and dim3 as discussed above) , observational speed (c ) and electric charge (e) .

It means that spinning mass and charge are intimately related to the number of dimensions it has to rotate through .

So , it is no surprise to find that

Beta = (1/10 + 1/27) * 10^3

=1000*(0.1 + 0.037037037…)

= 137 . 037 037 …

The difference in the sixth decimal can be attributed to drag effects and dimensional compensations , which have not been taken into account .

Biological Epigenetics .

The same type of argument can be applied to biological cells and denizens of multicellular organism . While they might not rotate , they definitely do partially rotate to-and-fro .

Three magnetic fields at right angles to each other or twistor-EM waves will have definite biological effects .

Do not try this at home .

Does nothing matter ?

The Zero knows .

Andre .

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Appendix A

From http://mathworld.wolfram/com/PrimeDiophantineEquations.html

From http://en.wikipedia.org/wiki/Formula_for_Primes

Formula based on a system of Diophantine equations

A system of 14 Diophantine equations in 26 variables can be used to obtain a Diophantine representation of the set of all primes. Jones et al. (1976) proved that a given number k + 2 is prime if and only if the following system of 14 Diophantine equations has a solution in the natural numbers:

α0 = wz + h + j − q = 0

α1 = (gk + 2g + k + 1)(h + j) + h − z = 0

α2 = 16(k + 1)3(k + 2)(n + 1)2 + 1 − f2 = 0

α3 = 2n + p + q + z − e = 0

α4 = e3(e + 2)(a + 1)2 + 1 − o2 = 0

α5 = (a2 − 1)y2 + 1 − x2 = 0

α6 = 16r2y4(a2 − 1) + 1 − u2 = 0

α7 = n + l + v − y = 0

α8 = (a2 − 1)l2 + 1 − m2 = 0

α9 = ai + k + 1 − l − i = 0

α10 = ((a + u2(u2 − a))2 − 1)(n + 4dy)2 + 1 − (x + cu)2 = 0

α11 = p + l(a − n − 1) + b(2an + 2a − n2 − 2n − 2) − m = 0

α12 = q + y(a − p − 1) + s(2ap + 2a − p2 − 2p − 2) − x = 0

α13 = z + pl(a − p) + t(2ap − p2 − 1) − pm = 0

The 14 equations α0, …, α13 can be used to produce a prime-generating polynomial inequality in 26 variables:

ie: PrimeNumber = (k+2) ( 1- a0^2 - … a13^2) )

This is equal to the polynomial

(k + 2)(1 −

[wz + h + j − q]2 −

[(gk + 2g + k + 1)(h + j) + h − z]2 −

[16(k + 1)3(k + 2)(n + 1)2 + 1 − f2]2 −

[2n + p + q + z − e]2 −

[e3(e + 2)(a + 1)2 + 1 − o2]2 −

[(a2 − 1)y2 + 1 − x2]2 −

[16r2y4(a2 − 1) + 1 − u2]2 −

[n + l + v − y]2 −

[(a2 − 1)l2 + 1 − m2]2 −

[ai + k + 1 − l − i]2 −

[((a + u2(u2 − a))2 − 1)(n + 4dy)2 + 1 − (x + cu)2]2 −

[p + l(a − n − 1) + b(2an + 2a − n2 − 2n − 2) − m]2 −

[q + y(a − p − 1) + s(2ap + 2a − p2 − 2p − 2) − x]2 −

[z + pl(a − p) + t(2ap − p2 − 1) − pm]2)

> 0

is a polynomial inequality in 26 variables, and the set of prime numbers is identical to the set of positive values taken on by this polynomial inequality as the variables a, b, …, z range over the nonnegative integers.

In other words , we have a single Diophantine polynomial equation with 27 variables based on 14 sub-equations .

Eliminating one variable (n) as discussed above , leaves us with 26 variables based on 13 equations , but the Exponential Order (Degree) is unchanged .

A general theorem of Matiyasevich says that if a set is defined by a system of Diophantine equations, it can also be defined by a system of Diophantine equations in only 9 variables. Hence, there is a prime-generating polynomial as above with only 10 variables. However, its degree is large (in the order of 1045). On the other hand, there also exists such a set of equations of degree only 4, but in 58 variables (Jones 1982). Jones et al 1976 , Riesel 1994 p40

Appendix B

Diophantine set

From Wikipedia, the free encyclopedia

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In mathematics, a Diophantine set of j -tuples of integers is a set S for which there is some polynomial with integer coefficients

f(n1, ..., nj, x1, ..., xk)

such that a tuple

(n1, ..., nj)

of integers is in S if and only if there exist some (non-negative) [1] integers

x1, ..., xk with

f(n1, ..., nj, x1, ..., xk) = 0.

Such a polynomial equation over the integers is called a Diophantine equation. In other words, a Diophantine set is a set of the form

where f is a polynomial function with integer coefficients. [2]

Matiyasevich's theorem, published in 1970, states that a set of integers is Diophantine if and only if it is recursively enumerable. A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever. This means that the concept of general Diophantine set, apparently belonging to number theory, can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work.

Matiyasevich's theorem effectively settled Hilbert's tenth problem. It implies that Hilbert's tenth problem is unsolvable. This problem is the challenge to find a general algorithm which can decide whether a given system of Diophantine equations has a solution among the integers. David Hilbert posed the problem in his celebrated list, from his 1900 address to the International Congress of Mathematicians.

Contents

[hide]

1 Examples

2 Matiyasevich's theorem

2.1 Proof technique

3 Application to Hilbert's Tenth problem

3.1 Logical structure

3.2 Refinements

4 Further applications

5 Footnotes

6 References

7 External links

[edit] Examples

The well known Pell equation

X^2 – d(y +1)^2 = +- 1

is an example of a Diophantine equation with a parameter. As has long been known, the equation has a solution in the unknowns x,y precisely when the parameter d is 0 or not a perfect square. In the present context, one says that this equation provides a Diophantine definition of the set

{0,2,3,5,6,7,8,10,...}

consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:

The equation a = (2x + 3)y defines the set of numbers that are not powers of 2.

The equation a = (x + 2)(y + 2) defines the set of numbers that are not prime numbers.

The equation a + x = b defines the set of pairs (a,b) such that (a<=b)

[edit] Matiyasevich's theorem

Matiyasevich's theorem says:

Every recursively enumerable set is Diophantine.

A set S of integers is recursively enumerable if there is an algorithm that behaves as follows: When given as input an integer n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0.

It is not hard to see that every Diophantine set is recursively enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0. Now we make an algorithm which simply tries all possible values for n, x1, ..., xk, in the increasing order of the sum of their absolute values, and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk.

[edit] Proof technique

Yuri Matiyasevich utilized an ingenious trick involving Fibonacci numbers in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam had shown that this suffices to show that every recursively enumerable set is Diophantine.

[edit] Application to Hilbert's Tenth problem

Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's theorem with a result discovered in the 1930s implies that a solution to Hilbert's tenth problem is impossible. The result discovered in the 1930s by several logicians can be stated by saying that some recursively enumerable sets are non-recursive. In this context, a set S of integers is called "recursive" if there is an algorithm that, when given as input an integer n, returns as output a correct yes-or-no answer to the question of whether n is a member of S. It follows that there are Diophantine equations which cannot be solved by any algorithm.

[edit] Logical structure

Here an argument taking exactly the form of an Aristotelian syllogism is of interest:

(Major premise): Some recursively enumerable sets are non-recursive.

(Minor premise): All recursively enumerable sets are Diophantine.

(Conclusion): Therefore some Diophantine sets are non-recursive.

The conclusion entails that Hilbert's 10th problem cannot be solved. The most difficult part of the argument is the proof of the minor premise, i.e. Matiyasevich's theorem, which itself is much stronger than the unsolvability of the Tenth Problem.

[edit] Refinements

Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the equation only has 9 natural number variables (Matiyasevich, 1977) or 11 integer variables (Zhi Wei Sun, 1992).

[edit] Further applications

Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable.

One can also derive the following stronger form of Gödel's first incompleteness theorem from Matiyasevich's result:

Corresponding to any given consistent axiomatization of number theory,[3] one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization.

[edit] Footnotes

^ The two definitions are equivalent. This can be proved using Lagrange's four-square theorem.

^ Note that one can also use a simultaneous system of Diophantine equations to define a Diophantine set, because the system

f1 =0 , …,fk =0

is equivalent to the single equation

f1^2 + f2^2 + … + fk^ = 0

^ More precisely, given a -formula representing the set of Gödel numbers of sentences which recursively axiomatize a consistent theory extending Robinson arithmetic.

[edit] References

Yuri Matiyasevich. "Enumerable sets are Diophantine." Doklady Akademii Nauk SSSR, 191, pp. 279-282, 1970. English translation in Soviet Mathematics. Doklady, vol. 11, no. 2, 1970.

M. Davis. "Hilbert's Tenth Problem is Unsolvable." American Mathematical Monthly 80, pp. 233-269, 1973.

Yuri Matiyasevich. Hilbert's 10th Problem Foreword by Martin Davis and Hilary Putnam, The MIT Press. ISBN 0-262-13295-8

Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Sci. China Ser. A, 35:3 (1992), pp. 257–269.

[edit] External links

Matiyasevich theorem on Scholarpedia.

Retrieved from "http://en.wikipedia.org/wiki/Diophantine_set"

Categories: Diophantine equations | Hilbert's problems

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