SRH - main

Returning to a past blog entry [ref here]. We may define a disk with a condition like

then we have a condition that cuts the real plane exactly. All points are either in the set or outside the set.

the complement

x^2 + y^2 > r where (x,y) is in R2

includes all other points.

While these two condition perfectly divide the real plane into two regions, there is no “evidence” of the division in the real plane. A being walking across the R2 would never know whether they were in the disk or outside the disk, only if they were subject to the same conditions as the mysterious meta rule above would they have any idea where the set boundary lay. This is exactly the experience of a human walking across the countryside and finding themselves having walked across a county or national boundary. The division doesn’t exist on the land only in rules of a meta language.

We may create an entity which represents symbolically the meta rule. Say a circle:

x^2 + y^2 = r where (x,y) is in R2

points within this circle are clearly separate from those outside. Yet the first example shows us that while we have constructed an actual set to represent our boundary it isn’t the boundary. The meta-rule have carved up the R2 into three sets now but there is nothing in the R2 which separated the point inside the circle from the points on the circle. That boundary doesn’t exist in the R2. There is an infinite regress as we try to create something which will act as a “real” constructed boundary.

In reality we may find a wall – like the Great Wall of China – which is hard to climb over or which is well defended. This creates a physical boundary. But a missile shot from a gun does damage to the body just as a rock falling from a cliff – the physical reality only represents the meta-laws it doesn’t construct them. Scaling the Great Wall of China would have given us access to the land on the other side – which is much like the land outside the wall. It is only a thought to say it gives access to China. Just as walking into the circle above crosses no actual boundary in R2, only a boundary as defined by the meta-rules.

So this is the old meta-language/ language distinction. It seems that it is impossible to construct anything. We begin with already existing entities (e.g. all the points in R2) and we divide them up based upon rules which are not expressible in those entities. Now this is the SRH.

A circle cannot be put in itself I suggest because a “circle” is a meta-construction – a set of points. While the set of points of a smaller circle are contained in the larger circle the smaller “circle” is only a symbol and can’t be “in” anything. This exaplins one thing but I need to use it to lever apart selfhood which I suggest is a two fold thing: the entity and the meta-language being confused in 1 entity.

This is back to AIME97 – is there a way of encoding the above rule into the R2? Make it easy.

{x | x>=2 & x in R}

so we have the points on the Real number line above and equal to 2 separated from less than 2.

1) Find the Turing machine that does this (tho it won’t work on the Real numbers which is non-computable!!!)
2) Is there such a thing as a “flat” Turing machine that stores its state in the Solution space without changing the solution!

=== but before that however consider…

In reality we don’t begin with a rule to make a circle. We have more this type of arrangement as see in the picture.

This way there is an “existing” and “constructed” entity. If we wish to create a rule to describe the entity we may approximate to a circle and describe the band of the boundary in the same way as above. From the standpoint of the rule it simply defines which points in the plane are in the set of the boundary. As above the rule does not represent any”thing” in the plane – it is not constructed. A being walking across this plane would encounter the black circle but would never know that they were walking across boundaries that are meta-described by the rule.

It seems that by rules alone we cannot produce an entity in the plane. The circle that we see can only be described by the rules – its “existence” or construction in the plane is a mystery from the standpoint of rules. The first one was created in a paint package.

This next one was created in a programming language. It is seemingly “constructed” from rules. Pixels were set according to the same rule as above. The first thing to notice is that it is not in the Real numbers but in the integers – the pixels are discrete entities. Unlike the general rule the computer scans through discrete angles and sets pixels one by one. All the hardware, the graphics memory and the screen pixels already exist. All the rule does is determine what colour to set a pixel. Nothing is actually created; the rule simply separated things that already exist. The rule doesn’t itself exist.

If this continues that rules can’t “exist” then if we postulate the existence of something then it can’t be the rule of something else… least of all itself.

This is getting very close to Russell and Whitehead (how boring).

Define a universal set E = {x/2 | 0< x <=10 & x in Integers} = {0.5,1,1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5}

So we can create subsets of E like

A = {x is an Integer & x is even & x is in E} = {2,4}
B = {x is an Integer & x is odd & x is in E} = {1,3,5}
C = {x is not an Integer & x is in E} = {0.5, 1.5, 2.5, 3.5, 4.5}

Now we can create a set of all sets in E

D = {A,B,C}

D is now a set in E so actually a full set of sets is like this.

D = {A,B,C,D}

and it is a very short step to creating the set:

F = {all sets in E which do not contain themselves} = {A, B, C}

so F does not contain itself so F should be {A,B,C,F} which does contain itself and the set is un-decidable.

But before we get here the SRH has a problem with D, but why?

In light of the above I want to ask whether A,B or C are “in” E. Remember the rule that divided the points of R2 into the disk did not manifest “in” R2. So why should A,B or C manifest in E?

A,B,C are “collections of numbers” and so are not numbers themselves and do not exist in E which was defined above.

So where do these “collections” exist? Where does the disk rule above exist? The SRH maintains that the space which is governed by a rule cannot contain the rule… thus the Horatio or God principle played around with before… to create a rule over a space one needs to construct a new space inaccessible to the first space.

BUT Godel did exactly this… he showed that Natural number space (N) was governed by rules in Peano Logic (P) but that these rules were isomorphic with numbers through Godel numbering. In the example above this is like encoding the disk condition into pixels.

BUT my contention here is that a new space is needed to govern Godel numbering. The isomorphism doesn’t “exist” in either Natural numbers or Peano Logic but in Godel numbering which was added (as it’s name suggests) by Godel. The rules that establish the correspondence between numbers and strings of logic are not evident in either Logic strings or Natural Numbers any more than the being walking across the disk in R2 is aware of the condition that separates the points in R2.

Godel creates a new space to establish the correspondence. Now this means that entities in N (numbers) do not “exist” in P (logic statements) and vice versa and so they do not “exist” together any more than the points in the disk are a rule and the rule that separates them is a point.

2DO: Introduce a Godel space however and I have my way of plotting the rule! Excellent. AIME97 is alive again… create a small logic that enables all points in N2 to map to a meaningful rule which in turn determines which pixels to select…

OK it is late… it is exam week and I’m doing SRH a bit prematurely… should wait till next week when I am 100% free again…