vol III Development:
Chapter 2: Model
page 3: Immensity: the transfinite symmetric universe
Mathematics
Traditionally, God is infinite. To talk about God, therefore, we need a language capable of dealing precisely with infinity. To get such a language, we must augment natural language with mathematics. Aquinas
Mathematics begins from natural languages like, as here, English. Nowak et al note that
Animal communication is typically non-syntactic, which means that signals refer to whole situations. Human language is syntactic, and signals consist of discrete components that have their own meaning. Syntax is requisite for taking advantage of combinatorics, that is 'making infinite use of finite means'. Nowak et al
Mathematics goes further, making transfinite use of infinite means. The infinite means are the natural numbers, 0, 1, 2, 3 . . . . The properties of these numbers are summarized by the Peano axioms. Beginning with zero, each new number is generated by adding one to its predecessor. There is no last natural number. Even though every natural number is finite, the set of all the natural numbers is infinite, in the language of set theory 'countably infinite'. Peano axioms - Wikipedia.
Set theory was founded by Georg Cantor toward the end of the nineteenth century. Because of its clarity and simplicity, set theory has become an important way to represent mathematical ideas. Although it has been formalized and vastly extended since Cantor's time, we will follow Cantor's 'naive' approach. Cantor, Jech.
Cantor defined a set (or 'aggregate') as 'any collection into a whole S of definite and separate objects s
of our intuition or our thought'. These objects are called the elements of S. Symbolically, we write
So we write the set of natural numbers N = {0, 1, 2, 3, . . . }.
Cantor defines the 'power' or 'cardinal number' of a set S as 'the general concept, which, by means of our active faculty of thought, arises from the set S when we make abstraction of the nature of the various elements s and the order in which they are given.' Cantor, 86 That is we consider the set as simply a collection of units of no specific kind.
Since there are an infinite number of natural numbers, the cardinal number of N cannot be represented by natural number. Instead Cantor uses the first letter of the Hebrew alphabet, aleph, and we write card(N) = ℵ0. (Cantor, page 103)
Order
The natural numbers described by the Peano Axioms have a natural order which arises from the process of their generation, so that N is an ordered set. From an ordered set we derive the notion 'ordinal type' 'which is itself an ordered set whose elements are units which have the same order of precedence amongst one another as the corresponding elements of S, from which they are derived by abstraction.' (112)
Cantor notes that
'the concept of ordinal type developed here, when it is transferred in like manner to multiply ordered aggregates embraces, in conjunction with the concept of cardinal number or power . . . everything capable of being numbered (Anzahlmässige ) that is thinkable, and in this sense cannot be further generalized. page 117
Here I understand thinkable to mean consistent, something that fits together seamlessly. Sometimes appearances can be deceiving, as with many of Mauritz Escher's paradoxical two dimensional representations of three dimensional spaces. M. C. Escher Company B. V.
Following the lead of natural language, we can make larger numbers out of the natural numbers by combinations and permutations. Given the set {1, 2, 3} we can produce the following six permutations: <1, 2, 3>, <1, 3, 2>, <2, 1, 3>, <2, 3, 1>, <3,1,2>, <3,2,1>. A set of cardinal number four can be arranged in 24 different permutations, and in general, a set whose cardinal is n can be ordered in n ! (n factorial) = n x (n-1) x . . . x 2 x 1 different ways. The size of n! grows exponentially with n, as Stirling's formula shows. Stirling's approximation - Wikipedia
The permutations of the natural numbers N may be considered the elements of a set with an even greater transfinite cardinal, ℵ1. Since these permutations have a natural ('dictionary') ordering, we may consider the permutations of these permutations to arrive at a set with cardinal number ℵ2. This construction of larger and larger ordered sets may continue without end. Like the natural numbers, there is no last transfinite number.
The generation of transfinite numbers seems to be closely related to the functioning of imagination. The proces of imagination like building with blocks, is a continual exploration by combination and permutation of the relationships of different elements, the characters in a drama, for instance.
We may call the sequence of transfinite numbers envisaged as a recursive process of permutation the symmetric Universe, by analogy with the symmetric groups constructed by permutation. Because there is no upper limit to the size of transfinite numbers, we can be assured that there will be enough transfinite numbers to form a one to one correspondence with the fixed points in the Universe no matter how many fixed points there are in the divine Universe.
We may consider the observable fixed points in the Universe as revelation of the seamless dynamics of God. The correspondence between these fixed points and the transfinite numbers serves as a measure of the immensity of the divinity. Our modern understanding the of Universe, with its vast spaces and huge numbers of particles, ranging from fundamental particles to planets, stars and galaxies is an observable representation of this immensity.
(revised 15 August 2014)
