Let us invent an "intelligent agent" an IA. Put them in a sealed box with a letter box to feed information. Load this onto a truck and drive it somewhere, before setting it down again.
Imagine this box is in a field. Take a photo of the box in the field and post through the letter box.
The IA receiving the photo, and wanting to know where they are, may assume that the box in the photo is their box. They may then deduce that they are in a field. But this only works on the assumption that the box in the photo refers to the box they are in. They do not actually know for sure whether this is not just a photo of another box. However this illustrates the basic SRH argument that a "structure" that contains an isomorphism with itself (a picture of itself) also contains an isomorphism of what is not itself, and what is therefore greater than itself (the field in which the box lies). So if we complete the identity and make the correspondence between the entity and its isomorphism, then we can make this deduction.
Note that the process of making this correspondence lies outside the "structure." A structure cannot represent itself and the iso-morphism between itself and its representation. Suppose the box was transparent and the photo was really a tablet so that the image from the camera was being sent wirelessly in real time to the IA. They would see an IA looking at a tablet in the box, and on the face of the tablet would be the image they were looking at. They might also recognise the IA as themselves. It seems like the correspondence between the box and the image of the box on the tablet is being encoded within the image. However you could create this image like a fractal and really the box is on a beach. So even if the IA realised that indeed they are looking at a tablet on which is a picture of themselves in a field in a transparent box looking at a tablet, they cannot reliably deduce that they are really in a field. Indeed in the style of the film "Matrix" they might even be having this image transmitted to their brain and they are not even in box and looking at a tablet.
Matrix Caveat: however Matrix is wrong about one thing: you would remember the world in which you learned to speak and live. If indeed you learned to live and speak in the Matrix then actually the Matrix is reality, and the red pill actually takes you into a new reality. Matrix tries to suggest the true reality is the world of people being farmed by machines. For this to be true peoples' upbringing would have to be done in this world, and they would have a memory and legend of being captured by machines and plugged into the Matrix. You cannot just rewrite the whole of history because our very language and thought processes are historical. This is the weakness of this type of thinking.
But once the identity is made either by machine or by the mind it is recursive. Because looking at the photo of the box the IA understands that inside it is itself looking at the photo. This enters an infinite regress like when a camera outputs to the screen it is pointing at.
In theory this regress is also an infinite progression going the other way. But in the real world there is a time delay to process and display the image so to go backwards would involve going back in time.
This raises questions about what "reality" is in fact. A computer can simulate itself. You can emulate anything. So if an Intel chip emulates itself, it can run programs that were written for the chip, on the software, which is running on the chip. So you can load the emulator, into an emulator. You can do this to any depth. Once you have some depth you can unpick it, this is running on a emulator, which is running on an emulator.. all the way up to this is running on a real chip and then the progression stops. Why? What is the difference between a simulation and the real thing? In this case the emulator only emulates the chip logic, it doesn't emulate the electronics. The emulator can run on the chip, but the chip cannot run on the emulator. Instead the chip is run on physics with cleverly designed electronic circuits that simulate logic. You could write a physics simulator to simulate the chip. Then you run the physics simulator on the simulated chip. But in reality physics starts to become very non-discrete as you get into more detail. Before long you would be simulating quantum processes using a random number generator. Indeed perhaps the quantum world has to be non-discrete in order to close off the possibly of infinite progress. So sooner of later in the logical world you must have a top to the simulator stack: a mother machine on which things ultimately run.
So identity is impossible to prove in information theory. You can never be 100% sure that the accused really did the crime. It might always be someone else who is either protected, or who very cleverly set up the defendant. But in physical systems you can do this. Consider a crane lifting a large metal plate. The system works. But now move the crane so that it stands on the metal plate and try and lift it. Suddenly the crane's motor is not longer using the crane structure to fight against the force of gravity, instead the crane motor is fighting the cranes structure: the crane is fighting itself. Probably either the motor will fail or the structure will fail, but we can be 100% sure the plate will not lift. The reason is to do with SRH. What the crane does is take its stable foundation and transfer that rigidity to the plate which is only linked to the ground by gravity and so it lifts it. However once the plate is connected to the crane they share the same stable foundation so there is nothing to transfer. A system relies upon what it is not in order to work. If it relies upon itself it will achieve nothing. This in spiritual terms in the problem with the ego. It is a mental construct which believes it can rely upon itself. Just like the crane lifting its own plinth, the ego can never achieve anything, and worse in trying to achieve things it puts the structure of the self under enormous strain. Worse still when things do happen, as they always must, the ego tries to claim that the source of the change came from within. All change comes from what is not, that is to say: what is outside the self. The self can only ever be just itself, it cannot change, it is impotent. But since it is not separate from the world, the world will change it whether it accepts this or not.
I have other things to do now but some progress. The SRH may not be able to prove God now, given that the proof relies upon a 100% certain identity between the things and themselves. But there are physical systems that certain do seem to have 100% correlation between themselves and themselves. So while the IA in the box looking at the tablet can never be quite sure whether this is a clever dynamic simulation of an IA in a box in a field, looking at a tablet, there is 100% no way that a crane can lift itself.
And, obviously in logic we can just define identity. This makes logic a different world from reality. When Godel numbers the statements in predicate calculus there is no doubt that each statement has a number associated with it, and the "computer" (in Turing's usage of human thinker) thinking through the proof can make certain correspondence and identity between numbers and statements. Though I note again: the original 2007 statement of the SRH was this: that the "computer" of a logical proof cannot be themselves the subject of the proof. Just like a crane lifting itself, were the proof itself to embody the "correspondence" then we would have no solid platform on which to make the identity between systems that is required for self-reference. In other words self-reference cannot be based on itself. There must be a system outside the proof (the computer for Turing) that thinks through the proof and upon which correspondence (identity) is formed. Without this Godel's Numbers are just ordinary numbers and have no correspondence with statements at all.
n ∈ K ≡ ~(Bew [R(n); n])
Only works if we, the Computer, interpret R(n) ... and I need read that proof again to be sure this is correct.
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