Proof of God progress

If a(b(x)) = f(x) and b(a(x)) != f(x) then there is no a(x) nor b(x) such that a(x) = f(x) or b(x) = f(x)

Basically the existence of "meta information" i.e. order of operations is outside either operation a or b. Thus a() and b() cannot be equal to f(x)

This is a concrete example of the SRH which states that self reference is impossible. If self-reference means more than simply self-naming which is trivial, but a computational function e.g. with self-knowledge or self-consciousness, then we are saying that bcos the Total system has components that are structured then the components cannot be both components in a structure and also represent that structure. Structure is a concrete example of the more general "meta" system. Originally it was awareness that text itself is meaningless, it needs a language user.

Equally computer code is meaningless it needs a CPU. Thus the CPU can never be expressed in code. However emulators exist, and emulators of Turing capability can emulate anything. But at root the emulator must run on a CPU that is not itself emulated. You cannot have M.C.Eshers hand drawing a hand drawing itself, it must have basis outside the system. This is because the emulated entities while homomorphic with the "real" CPU can be "necessary components", and thus there must be the possibility of information that is outside the emulations.

Definition: "necessary components" are components whose structure is critical to their compositions of the entity. A grain of sand in a bucket of concrete is not a "necessary component" as the grain of and can be swapped with a pebble and it is still a bucket of concrete. There is no structural information. However a strut in a roof frame is. The roof frame fails if the strut and joist components are swapped.

There is an exception to the opening statement.

While a(b(x)) != b(a(x)) across most values, there may be a Fixed Point value.

Since a(b(fp)) == b(a(fp))

Fixed Points were central to earlier exploration of the SRH. Godel Statements lead to contradictions and so incompleteness which breaks systems and allowed statements to exist "outside" systems, or the boundaries of systems to be broken. This is SRH, in that systems are forced to acknowledge an "outside". A truly self-referential statement would reference itself within itself, that would be a statement that would need no "outside", one that depended only upon itself. "Myself" is synonymous with totality and exclusion of "Other". This is very much the Dark-Tetrad problem discussed in previous post.

The negation of a tautological statement is a contradiction.

Similarly it was theorised that where a fixed point in a system like: x = "x is True" cannot be false without contradition, its negation x = Not("x is True") is a contradiction and Godel uses this structure to break Principia. x = Not(x is in Principia)

This is very loose logic, probably a lot of nonsense, but broadly sweeping out over the mental possibilities.

So it is interesting that Fixed Points that were the focus before, and which we thought provided the mechanism to break any system and point "outside" defeating any Totalising project like Russells. Now they are the unique cases that break my proof that meta-data, or structure must exist in the order of operations that cannot therefore be grasped by those operations.

If we rewrite f(x) == a(b(x)) as f(x) == {a,b,ab}(x) meaning that we have a, b and the order ab.

If A(x) == {a, ab}(x) i.e. somehow it was able to contain the structural info so that when combined with b there is "nothing more" and it is then equal to f(x) so that:

f(x) = A(b(x)) = b(A(x))

we would need to show that whatever {a, ab}(x) represents is always a chimera.

Tentative steps forward. It is encouraging that this looks like nonsense. Either it is madness, or it is pushing against something (to me) new.