##### volume **II:** Synopsis

#### Part II: A brief history of dynamics

### page 16: Georg Cantor

(1845-1918)

The natural numbers, 1, 2, 3 . .. are infinite, since we can always add another one. It was known in antiquity that Pythagoras' theorem implies that there are quantities that cannot be measured by the natural numbers or by the rational numbers, ratios (eg 3/5) of natural numbers. To measure such quantities, we must invent the 'irrational' or real numbers. Cantor showed that the step from natural to real numbers is not unique, but the first an endless series of steps to even bigger number spaces, which he called the transfinite numbers. Following Cantor, our natural theology is based on the hypothesis that the transfinite can help us to understand God. Incommensurable magnitudes - Wikipedia, Hallett

One might argue that the oldest scientific problem in the world is the relationship between discrete and continuous systems. A continuous system is something like a wheel, which moves smoothly without jumps or gaps. A discrete system is the exact opposite. Each unit of the system is separate from the others, and if one is to move, one must jump the gaps between the elements of the system. Language is a discrete system. Different words and sentences have a gap between them.

Aristotle, and many others, felt that the existence of motion meant that space must be continuous. The idea of continuity is closely connected to the idea of infinity: one may think of a continuous line as an infinity of points; alternatively one may think of motion as indeterminate or infinite in the sense of not being definite. In the first point of view, the points are imagined to be dense, having no gaps between them. This idea seems to contradict the idea of a point, which something distinct, having, one feels, some sort of boundary between itself and the next thing. John L Bell

Cantor set out to find the 'cardinal of the continuum', that is the number of point it takes to make a continuous line. His attempt was later shown to be doomed to failure, but while searching he invented the theory of sets, which has become a foundation of mathematics. Cardinality of the continuum - Wikipedia, Set theory - Wikipedia, Cohen

Language is discrete, but it is also infinite. There is no limit to the number of new sentences that the speakers of a language can produce and understand. This infinity arises through the combination and permutation of elements of the language in accordance with the rules (if any) of its grammar. Essential to the notions of combination and permutation are the ideas of subset and order. Cantor used the notion of ordered set to construct a representation of the transfinite space of mathematical language. Nowak
The basic set of mathematical words is the natural numbers, 0, 1, 2, 3, . . . . These numbers are represented in natural languages by very different words and symbols, eg one, two, three; bir, iki, uc; etc, but we all know what they mean. Because the natural numbers are infinite, they provide us with an infinite vocabulary, and so an infinite domain of meanings. By meaning here we can understand a mapping between one symbol and another. Cantor imagined a set or class containing the natural numbers and called the number of numbers in this set *ℵ _{0}*, the first transfinite number. Cantor

He then went on to show that the infinite set of natural numbers may be assembled into a set of permutations which has a higher degree of infinity than the infinity of natural numbers. This next transfinite number he called *ℵ _{1}*. This, he thought, might be the cardinal of the continuum. Beyond this, we may permute these permutations to generate sets of even greater infinity, yielding sets with cardinal

*ℵ*and so on. The generation of the larger transfinite numbers depends upon ordering, analogous to the ordering of the elements of the Universe which gives complex entities, like ourselves. On this site we take the transfinite numbers to be a foundation on which to build a language big enough to begin talking about God.

_{2}, ℵ_{3}
Aristotle and Aquinas model God as pure act, *actus purus*. God is alive, that is an entity which moves itself, or, as mathematicians might say, maps onto itself. It can be proven that any complete formal system that maps onto itself has fixed points. The idea to be developed here is that Cantor's transfinite numbers are a representation of the fixed points in the pure dynamism of God. Aquinas 113: Is life properly attributed to God

[revised 23 May 2013.]