volume II: Synopsis
Section IV: Divine Dynamics
page 30: The transfinite network
We can imagine any organisation as a filing system and a set of processes for updating the files. This businesslike idea is here expanded, using Cantor's theory of transfinite cardinal and ordinal numbers, into an infinite abstract structure we call the transfinite network. We assemble a selection of the mathematical, physical and philosophical ideas developed so far into a model large enough to talk about God, but with sufficient finesse to deal with every detail of the world, no matter how small.
We are looking for a language or space to describe God. By God we mean here the Universe as a whole. We can only imagine two constraints on the nature of God. The first is consistency. God must be consistent with itself. The second is size. Cantor's theorem shows us that a consistent symbolic system as large as the natural numbers will grow without limit into the transfinite cardinal and ordinal numbers. This space, we postulate, is large enough to begin to model a divine Universe. Every point in it may be represented by a unique number, that is, a unique ordered set, or ordered set of ordered sets, of whatever symbols we choose to use to represent the natural numbers. Cantor's theorem - Wikipedia
Transitions from point to point are made by reordering sets, that is by permutations. Permutations are performed by computers. Turing, one of the inventors of the computer, envisaged two types of machine which he called a (for automatic) machines and c (for choice) machines. a-machines are deterministic, in the sense that once they are started they follow a definite course until they either halt (if their initial configuration is computable) or not (if the initial configuration is not computable). Alan Turing
c-machines, on the other hand, go through a certain number of steps until they come to a point where the next step is indeterminate and it must consult an outside agent to choose its next move. In modern terms, we might call c-machines network machines. One would expect an c-machine to be more powerful than an a-machine since it has outside help. The machines populating the transfinite network are c-machines. Most real computers are c-machines. This computer spends most of its time doing nothing, waiting for me to input another keystroke. I make the choices. Tanenbaum
Since the transfinite network is taken as a model of the whole Universe, every c-machine within it has, in principle, the ability to consult all the other machines, and can thus tap the power of the whole network. This is the first step in a recursive process, since we can imagine that once each machine has learnt everything all the others have to teach it, this new generation of more powerful machines can begin another round of consultation. And so on without end.
Turing pointed out that there are only a countable number of different automatic machines, corresponding to the countable set of computable functions. Computable functions are thus a relatively scarce commodity in the transfinite network, and like the limitation of resources which causes natural selection in the biological sphere, we might imagine there is competition for computable functions in the transfinite network. Natural selection - Wikipedia
One way to share this resource is through communication. We can imagine that there are many more c-machines than a-machines, since the c-machines explore the indeterminate halo predicted by Göde; to be surrounding the a-machines. By neworking, we increase the entropy and power of the information processing system. This phenomenon is not unique to humans, we postulate, but occurs at every level in the universal network. Kurt Gödel - Wikipedia
Although there are only a countable number of computable functions, there does not appear to be any constraint on the number of instances of these functions, integrated over space and time.
Although the number of computable functions is limited, these functions can nevertheless manipulate transfinite entities through the the establishment of correspondences between these entities (eg people) and the natural numbers. This is the way bureaucracies, national and corporate manipulate the people they control. All entities manipulate their environment to some extent to ensure their existence.
The power of an c-machine depends on its place in the recursive hierarchy of the network. The transfinite net has an infinite hierarchy, and so allows for the existence of machines of unlimited power, governing larger and larger subsets of the network. This abstract network provides us a huge space for modelling reality. Like all networks, it is layered, its lowest physical layer comprising instances of the initial singularity. From this structureless entity, we imagine the whole Universe being created by the network process of copying and differentiation. Computer network - Wikipedia
(revised 25 May 2013)