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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)

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Beale, R, and T Jackson, Neural Computing: An Introduction, Adam Hilger 1991 Jacket: '... starts from basics and goes on to cover all the most important approaches to the subject. ... The capabilities, advantages and disadvantages of each model are discussed as are possible applications of each. The relationship of the models developed to the brain and its functions are also explored.' 
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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' 
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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' 
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Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. ... It can be used as a textbook for a graduate course in set theory and can serve as a reference book.' 
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Tanenbaum, Andrew S, Computer Networks, Prentice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. ... This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.'  
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Links
Alan Turing On Computable Numbers, with an application to the Entscheidungsproblem 'The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.' back
Cantor's theorem - Wikipedia Cantor's theorem - Wikipedia, the free encyclopedia 'In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2n for all natural numbers n. But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite. The theorem is named for German mathematician Georg Cantor, who first stated and proved it.' back
Computer network - Wikipedia Computer network - Wikipediathe free encyclopedia 'A computer network, or simply a network, is a collection of computers and network hardware interconnected by communication channels that allow sharing of resources and information. . . . The best known computer network is the Internet. . . . Computer networking can be considered a branch of electrical engineering, telecommunications, computer science, information technology or computer engineering, since it relies upon the theoretical and practical application of the related disciplines.. back
Jeff Speaks Theories of Meaning (Stanford Encyclopedia of Philosophy) 'Here I focus on two sorts of “theory of meaning.” The first sort of theory—a semantic theory—is a theory which assigns semantic contents to expressions of a language. Approaches to semantics may be divided according to whether they assign propositions as the meanings of sentences and, if they do, what view they take of the nature of these propositions. The second sort of theory—a foundational theory of meaning—is a theory which states the facts in virtue of which expressions have the semantic contents that they have. Approaches to the foundational theory of meaning may be divided into theories which do, and theories which do not, explain the meanings of expressions of a language used by a group in terms of the contents of the mental states of members of that group.' back
Kurt Gödel - Wikipedia Kurt Gödel - Wikipedia, the free encyclopedia 'Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.' back
Natural selection - Wikipedia Natural selection - Wikipedia, the free encyclopedia 'Natural selection is the process by which favorable traits that are heritable become more common in successive generations of a population of reproducing organisms, and unfavorable traits that are heritable become less common. Natural selection acts on the phenotype, or the observable characteristics of an organism, such that individuals with favorable phenotypes are more likely to survive and reproduce than those with less favorable phenotypes. If these phenotypes have a genetic basis, then the genotype associated with the favorable phenotype will increase in frequency in the next generation. Over time, this process can result in adaptations that specialize organisms for particular ecological niches and may eventually result in the emergence of new species.' back

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