Quora answer: Kurt Gödel: What are the relations between Processes and Godel’s Proof?

Feb 16 2014

Processes and Systems are duals.

Systems and Meta-system schemas are inverse duals.

Godel Statements intervene to define the difference between inverse duals but do not intervene in the difference between duals.

If this is true, then that in itself is of some interest.

The difference between what I call the meta-system schema and the system schema formally is the difference between the Universal Turing Machine and the Turing Machine, which are isomorphic but which are still different in as much as Universal Turing machines are the operating systems for Turing Machines, and though they are identical in some respects they are very different in other respects, and one of those respects is that the Halting problem is no problem for Universal Turing Machines because as Operating Systems they are not expected to ever halt unless turned off. The Universal Turing Machine is the operating environment of the application turing machines it reads from its tape and runs. From a practical point of view Operating Systems are organized differently from the applications that they run, and they apply constraints to the applications that they run, and they impose protocols on them for them to communicate with each other, or with the Operating System. The operating system disposes resources to turing machines that it is running in parallel, or though task switching, or as single threads. So in effect the Meta-system and the System are in some sense the same yet in another sense very different and that is one of the reasons that meta-systems are invisible to us because we can treat them as merely ‘operating’ “systems” rather than recognizing the difference in their essence despite the sameness of their formalization. In effect the Turing Machine formalism is wider than just describing systems, it in effect describes two different schemas under the same umbrella, and because of this we have difficulty telling them apart. Another similar example is G. Spencer-Brown’s laws of form. It can be reduced to a Boolean algebra, but to say that it is nothing more than a Boolean algebra is a mistake, it ignores its emergent properties that overflow its minimal formal representation. Similarly the difference between the System and the Meta-system is that the System is a whole greater than the sum of its parts, and the Meta-system is a whole less that the sum of its parts, i.e. a whole full of holes. And this impies the existence of the third case, which are wholes exactly equal to the sum of their parts, i.e. the special systems. The difference between a system and a meta-system is that systems are emergent because they include the Godel Statements (about which decidability is impossible) while meta-systems do not include the Godel Statements. So it is the Godel Statements themselves that decide whether something is a system or a meta-system. The holes in the meta-system are the niches for the systems to fit into within it. Both systems and meta-systems self-reflexively nest within themselves each separately but with respect to each other they interleave like Russian Dolls where the dolls are the systems, and the spaces between them are the meta-systems. This self-reflexive interleaving is what Aczel calls a non-wellfounded set, i.e. it violates the law of Russell that classes not be members of themselves, but with mediation. In other words systems can contain systems but only mediated by meta-systems, and meta-systems can contain meta-systems but only mediated by systems, so that hierarchies of sub-systems, systems, and supersystems (systems of systems) have implicit meta-systems separating them and making what Bateson’ calls differences that make a difference. There is then what G. Spencer Brown calls a “mark” that distinguishes the sub-system from the system and that is the meta-system, and vice versa. There are the laws of form, but there are also the anti-laws of form which is its dual and these cancel. The laws of form exist within the system just as the laws of the system exist in the meta-system. In effect the laws imposed on a lower level schema come from the upper level schema in the cascade of schemas that are part of General Schemas Theory S1 hypothesis. Laws of Nature that are imposed upon us come from the environment of nature that encompasses us. Laws always come from the next level up and the fact that they are laws is part of the difference between the encompassing and the encompassed schemas that are adjacent in the hierarchy of schemas that include: facet, monad, pattern, form, system, meta-system (openscape), domain, world, kosmos and pluriverse. The Kosmos (universe) operates to the laws imposed by the Pluriverse (multiverse) and this is true all the way down. Each higher schema is the operating system for the lower schema. There are in fact turtles all the way down, but the levels of ontological scoping are finite. There are only ten levels of schema that go from negative one to the ninth dimension, and even though dimensions are infinite, schemas are not. Schemas are part of our finitude, because they are the way that we project spacetime on ontic phenomena we encounter as Kant divined. Godel’s proof plays a key role in the distinction between the system and the meta-system, and in fact every higher schema from its adjacent lower schema. It is always by the inclusion or exclusion of Godel Statements (that are undecidable) that these levels are distinguished from each other.

Systems are the dual of Processes, just as a gestalt is  the duals of a flow. These are complementary to proto-gestalts and proto-flows at the Meta-system level, that correspond conceptually to meta-systems and meta-processes. This difference is a duality, while the difference between meta-system and system or meta-process and process is an inverse duality. The Godel Statements do not condition to the differences between duals but only inverse duals. So the difference between a process and a system is not related to the difference between a turing machine and a universal turing machine. Systems and Processes are two different turing machines images. In one the state machine is primary and the tape or stack is secondary, while in the other the tape or stack is primary and the state machine is secondary. They are merely two arrangements of the same thing. In a gestalt the figure is on the background, but in a flow the background comes to the fore and is against the reference point which is the figure pushed to be background. These distinctions are not undecidable but merely complementary arrangements of the same elements.

This was meant to clarify distinctions between processes and systems and their relation to Godel’s proof which relates more to the difference between these duals and the meta-processes and meta-systems.

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