## Friday, September 28, 2007

### Topos 4

Topos 4
http://arxiv.org/abs/quant-ph/0703060 The topos articles.
http://.arxiv.org/abs/0704.0646

Andre Willers
28 Sept 2007

For Ermeine on request .
Don’t worry . We reach a natural pause soon .

Other Sources:
http://andreswhy.blogspot.com/ : see in general , or search for “mining the Oort” , “Topos ” , “Topos 2” , “ Topos 3 “ etc Transcendent numbers ,etc.

First , a Godel detour .

Arith II :Defined as
The standard axiom-set of 10 axioms that gives rise to our set of numbers .

I here use the usual number-line as learned at school . A line that starts at 0 and continued indefinitely (to infinity) via the property that (x+1) <> x .
Ie that numberline never loops back on itself .
But this axiom is arbitrary .
(The axiom that gives difficulty is usually given as the axiom Number 4)
This is analogous to the parallel-axiom in Euclidean systems that gave rise to non-Euclidean geometry .

Godel proved rigorously that if you have a theory that incorporates ArithII , then you can formulate statements which cannot be derived (proved) from the given axiom set as True or False unless you incorporate at least one extra axiom . In other words , at the heart of our number system lies statements which are both True and Not-True .

This is equivalent to stating that the set of statements represented below cannot be proven complete unless n is an element of ArithII .

But if n is an element of ArithII , then it is infinite . Not just an ordinary infinity , but an infinity where aleph -> omega .

ArithII + A(1) = Arith I(1)
ArithII + A(2) = Arith I(2)
.
ArithII + A(n) = Arith I(n)

Where A(n) is an axiom defining a loopback of the numberline .
As you can visualize , the loopback can terminate on any point of the ArithII numberline . Hence , n must be an element of ArithII .

This is not a very rigorous proof , but true nonetheless .

What does it mean ?
1. No General Unified Theory (GUT) is possible using a finite set of axioms .
This means that mathematics and physics are open-ended . There is no end to new understanding (axioms) . Literally .

2. An entity capable of using infinite sets of axioms would be a Reality Class four entity . Ie strong godlike .

3 . Local groupings of axioms .
These can be found by algorithmic means . At least two rigorous ones and one iffy one are known :

3.1 The Scientific Method
3.2 Transcendand number tunneling (ie numbers like e , pi) . (Note my use of 1/e in previous posts.)
3.3 Direct inspection . (See “Souls” below)

The Willers-Venn diagram .
Ordinary set-theory cannot be used , since elements in the Universum cannot be given a Defineability Axiom .

This is the trick :
Use a window on the Universum . Simple ,eh? But very powerful .

Take a piece of paper and draw a rectangle .
This is a window on the Universum , all the defineable and non-defineable thingies .

Now you do not have to worry that the Universum has no boundaries , as long as your internal Venn diagrams do not touch the sides of the window .

Inside the window , draw the normal intersecting Venn circular diagrams denoting Y and N and not touching the sides of the window . The Intersection being YN .

It is not easy to find YN by just looking at Y or N .

But we can sneak up on it by looking at Not-Y and Not-N .

In Aristotelian systems ,
{Not-(Not-Y)} Intersection {Not-(Not-N)} = YN

but in Non-Aristotelian systems
{Not-(Not-Y)} Intersection {Not-(Not-N)} = YN + Everything outside the Y or N circles .

Conservation Laws and chaos .
If you have a system where nothing has been defined , and no conservation of any kind can be found , regardless of defineability and repetitiveness axioms , then the system can be described as truly chaotic . No internal space or time metrics can be found . In other words , it is a point . But in an infinite sea of such universes , some will be closed and some will interact . The interacting ones give rise to the phenomenon known as vacuum energy .

There is a further discontinuous spread of interaction giving rise to “particles” .

The essential point is that if there is a conservation law in the set Y , then there is a mirrored conservation law in the set N . Most important that there is then a conservation law that can be found in the “pseudoset” ( YN + Everything outside the Y or N )

This means that elements of YN is holographically spread all over the show , regardless of any space , time or any other constraints less than class 4 reality .

Does this remind you of anything ?

Souls have never been proved or disproved after 70 000 years of trying .
Yet they are as real as a car-bomb in the market .

The conservation effect in YN structures means that souls do exist . But our definition of individuals as a traveling standing-wave of memories needs some work , as each wave terminates on going to sleep . The waking is rather a random affair . See previous posts .

Can the soul be destroyed ?
Yes . If a class 4 entity “looks” at you , YN->0 and the elements of personality become fixed . Free will , which is another description of personality , ceases . Of course , the entity can restore the YN state vectors exactly . But why ? If He bothers to Look , you have the Heisenberg Hop with a vengeance .

Class 3 entities or less cannot destroy a soul . Attempts to do so by restricting the freedom parameters of individuals end in tears .This is true regardless .

The inverse is interesting . Increasing the freedom parameters of individuals usually works . but it can badly wrong as with Nazism and Stalinism . Then it results in even larger scale bloodletting .

Ah well .
The rest you will have to work out for yourselves .

Andre