SRH : +1 phenomenon (again)

If we have n full boxes which can only hold 1 item, we need an extra box to enable a swap operation.

This is a great example of the SRH. The system is so defined that it is "full" and "swap" is so defined that it cannot be completed in the full system. The system needs to expand to enable swapping.

It means that no "full" system can have a "swap" operation defined on it!

This gives us some grip on the idea of a systems "fullness".

This is vaguely akin to the pigeonhole principle. Consider a old style company with 10 employees and so 10 pigeon holes for internal mail. If we learn that 11 items got delivered today then we can deduce for certain that someone received at least 2 items. In the most egalitarian situation where everyone got the lowest amount of mail, all the pigeon holes would be full and so the 11th item would have to be put in a hole with an existing item. At the other extreme someone could receive all the mail. Either way someone has 2 or more. 

This is also similar to the SRH which remains still poorly defined, that a system cannot represent itself because there is "not room".

Self-Reference Revision
There are 2 types of self reference.
(1) Naming
(2) Isomorphism

(1) In Naming we simply identify an object by a name. Then we can reference that object with the name. Thus I can speak of the "Statute of Liberty" and "Unicorn" in the same sentence without any regard what I am talking about.

Predicate logic is so defined to avoid presupposing ontology. So we have sentence like "Ex | x is a unicorn". This is false. "-Ex | x is a unicorn" is true. Which says in English there is no x such that x is a unicorn, or more naturally there is no thing which is also a unicorn. Here we have no knowledge of x other than whether the qualifier selects it.

SRH does not apply to naming. If we are able to list all the statements in a language in order we can generate a Diagonalisation function #(x) which generates the number for each statement. This number list is then isomorphic with the entire calculus and statements can refer to themselves using these numbers. But using numbers is just naming. The actual self-reference lies in the isomorphism between the list of numbers and the list of statements. And this depends upon the Diagonalisation function. This is the important 2nd type of reference. According to SRH then the Diagonalisation function is the gremlin in the system that breaks logic. Is there controversy about whether Godel is correct that such a function can be created? I imagine by now someone would have provided the contradiction if not.

??Note to self: Fixed Point Theorem

So in Full Self-Reference a system must map to itself, be isomorphic with itself. This is like someone finding themselves on a map. The map is isomorphic with the landscape, so when it is opened in the area is represents the user can put a pin in where they actually are. There is in fact a point on the map that sits right over the point on the ground it represents! For this to work the map does not contain every detail just enough for the user to work out a rough area. However in the world of maths and the odd behaviour of infinity it is quite possible for an object to contain itself. However important not to get carried away. When an object contains itself it must repeat. Consider these arguments about the sequence Pi .

Everything can be expressed as a number sequence. In the modern age of digital information everything becomes just sequences of {0,1} which is just a binary number. Is there anything that cannot be stored on disk? If it can be measured then it can be stored. Obviously an elephant can't be stored on disk, but all information about an elephant can. And its quite a philosophical question to ask what an elephant is if its not the information about it? We can dispense with Noumena immediately. But that's something for a later post.

So the question of full self reference is isomorphic itself (or maps) onto the question of sequences containing themselves. As the arguments above show irrational numbers (which are by far the most numerous numbers) are unique sequences/strings which do not contain other irrational numbers. Any number that contains itself repeats. There is an issue here with what Mathematica calls "Flattening." The sequence 123412341234... repeats after 4 digits but if we say it contains itself it is like this:
123412341234... (object x)
____123412341234... (a copy of x maps onto x at position 4 - starting index at 0)

But we have relative offsets and absolute offsets. So the sequence offset by 4 maps into itself at offset 4, which is offset 8 in the original identity sequence. There is an exponential tree of offsets each creating a new object and so new relative base. These can all be flattened to an absolute index. I do not know if this is important, but when we say "maps onto itself" an infinite number of answers apply and these can be in relative and absolute depending which "self" we are talking about. Ignoring for now but it illustrates the literal "hall of mirrors" when things reflect themselves.

So returning to the top. Systems do have sizes. Pi is not an infinite space despite being an infinite sequence of numbers. It contains the sequence of numbers that is Pi and Pi only. It is contained itself it would repeat and so be rational and expressible as a fraction. It is proven irrational. It cannot also contain the sequence of any other irrational number. There are an uncountable number of sequences that it does not contain! (I need find proof of this).

So the idea of "full self reference" does seem to be in trouble as isomorphism with oneself means that you must be expressible as a fraction. It is that repeating that creates the redundancy and "space" in order to perform self isomorphism.

This also suggests that like swapping, self-reference is a function that requires "room". It is not a transparent property that sits on top of a system without taking up discrete "space" within the function.

As fractals demonstrate the closing of a system by feeding the output into the input generates constraints around fixed points. It is these fixed points like the repeating in a self-isomorphic sequence that limits the system and creates the repeating patterns and constraints. Closed, self-referring systems are not free. They internally define constraints and structure. The whole idea of SRH was to prove God, or at least end the idea of Full Knowledge. That the unknown is an essential component of known. Indeed to know something proves there must be things you do not know. If we were to know everything, then knowledge would be isomorphic with itself. The binary sequence of all knowledge would have to be isomorphic with itself, otherwise its own sequence would present new knowledge that was not known. But to know everything would place rigid constraints on the "space" in the system of knowledge.

Some people like Wolfram argue that this internal structure is isomorphic with the structure we see in the universe. The universe is a fractal generated around fixed points within the rules of dynamic physical systems. That is the positive take. The negative take is that there are immovable structures generated by self-reference.

The interest in fixed-points earlier in this blog arises because these structures that are created by self-reference certainly puts self-reference at the centre of the world (and perhaps consciousness) but can these structures that are generated around the fixed points of the system be isomorphic with the system? The Mandelbrot set famously is self-similar but each copy is different and occurs at arbitrary locations. If we can show that any of these locations are irrational then even the sequence of co-ordinates of repeats cannot be contained in itself so the Mandelbrot set is definitely not self isomorphic. This means that the set provides information that is not present in itself.

I would guess then that if something is self-isomorphic then the fixed points are new information which is outside the system, beyond the space of the system. And if something is not self-isomorphic then it itself is extra to the system. Either way no body of information can be complete. There no limit to knowledge. God is safe in the awareness that knowledge is bounded.

That's the best account of SRH so far. Perhaps getting there.