SRH inside/outside

So the SRH originates from ideas like inside/outside.

Are we confused? There are 2 formations:

A) a system isn't "big enough" to encode itself within itself. A crude reading of this leads to the Quine which apparently can produce itself. But a Quine must be "read" or "run"; it exists in a context. To take care of this suppose we have a Quine in ARM logic. We still need to know the ARM chip to make sense of the code. Perhaps the best quine would output an ARM emulator that runs on an ARM chip, with a hardcoded memory containing a quine that outputs the whole thing. When run on an ARM chip it emulates an ARM chip and runs its memory to create the source machine state. This way the Quine is at least isomorphic with the hardware. But the machine state and the emulator state are not identical. The machine is running an emulator, while the emulator is running a quine. I'll need to think whether there is a way out of this. But the point here is that we are saying that a true self-representation, a true mirror, is impossible because you can't mirror the mirror itself. There cannot be complete isomorphism between the system in situ and the system encoded because the "in situ" is too big.  

B) a system with self-reference must code an outside the "+1", "Horatio" formulations of the SRH.

Actually isn't this Godel's 2 theorems. "A" is the first theorem saying that if you try to take the whole system you will fail to account for everything. And "B" is the second theorem saying that if you have a system with self-reference it will create a contradiction.

Anyway the point of this post was to note that all these issues can be encoded at root with just a binary distinction. The inside is the 0 and the outside is the 1.

When Tarski argues that Truth and Falsehood cannot be encoded within a logic because then you have the statement "Sentence n is false." If you list all the possible statements (Diagonalisation) so that each has a number than this statement becomes a problem. What does this statement say about the 'n' that is its own number in the list? It says that this statement is false. Its the classic Liar Paradox. Essentially the problem lies in trying to order ALL the statements into a binary classification. Russel's Paradox is the same, and Godel's Theorem is exactly the same: Provable and Non-Provable (function Beq). When we take a language that enables self-reference even obliquely like with Diagonalisation (e.g. Godel Numbering) then we have a problem splitting it in half so that the system is isomorphic with the set {0,1}. By oblique "self-reference" we mean we have an isomorphism between the statements of that language and the entities that those statements are about x (x is always natural numbers) so that we can say things about numbers which are understood to also be the position of statements in a list, thereby making oblique reference to oneself e.g. "I am sentence 5".

At its simplest level the SRH lies in the problem of encoding the distinction between {0,1} in a system itself. A system simply cannot be perfectly divided, some parts will always fail to classify under any decision algorithm. Or if we succeed in dividing it then the system becomes isomorphic with the 1 and so the 0 implies that there is more (an outside) to the system which isn't encoded.

As an aside the kind of systems we are talking about are things like a map being placed within the territory that is isomorphic with the map. It means we can put a point on the map that corresponds with the position of the map itself. Such self-reference is what causes all the problems and why SRH is called "Self-Reference Hypothesis" although we do not yet know what that hypothesis is.

Its a bit like Hofstadter's observation that self-reference always seems to be self limiting. One needs to be very careful with such statements as like with Quine's which seems impossible at first glance turns out to have a trick. But once we have isomorphism with oneself and we introduce a division then this division can be applied to oneself and this is where all the internal contradictions come from. This is what Godel really showed. But going the other way up the tree, once we have a distinction and we have self isomorphism then we can show that the division must hold outside the system so the system becomes a part of a greater division. We can put the real position of the map on the map, or we can look at the position of the map and see what lies outside the map in the world. The difference between working out where you are on a map from the surrounding features, and finding the "you are here" arrow and then working out what is around you. In fact the 2nd formulation of SRH I will now term "You are Here" this is the "+1", "Horatio" or "Proof of God" versions previous mentioned in the blog. The "Proof of God" comes from the ability to always induce a greater context to any system with self-reference. God in Godel language are those statements that cannot be proven with the existing axioms. All systems have an "outside." Interesting that Godel himself searched for a proof of God but never accepted this as proof?

So indeed SRH does seem to have 2 versions. The "inner contradictions" from applying the divisions within oneself, and the "outer implications" from induction into the world outside the system.

On one level there is the search for a solid proof or at least formal description of this whole family of issues.

On another level suppose we already have such a formulation of the SRH. What would it say about itself? As mentioned before in this blog is there a fundamental problem with the SRH itself?