vol 2: Synopsis
part II: A brief history of dynamics
page 16: Georg Cantor
Site map
Directory
Search this site
Home
1: About
2: Synopsis
3: Development
Next: page 17: Einstein
Previous: page 15: Analysis
... to restore theology to the mainstream of science
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 theology is based on the hypothesis that the transfinite can help us to understand God. 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, since one may think of a continuous line as an infinity of points. These 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.
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.
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.
Cantor showed that the infinite set of natural numbers may be assembled into a set of sentences which has a higher degree of infinity than the infinity of natural numbers. Beyond this, we may assemble these sentences into a set of texts of even greater infinity, and so on. This structure, called the transfinite numbers, we take to be a language big enough to begin talking about god.
Later we will see how this transfinite structure has been used by physicists and mathematicians to describe the universe. Our claim that the universe is divine is partly based on cardinal numbers: physicists have found that only the transfinite space is big enough to describe our world in complete detail. The universe so revealed is not the finite result of a creative act, but the infinite result of self creation.
Books
| Bernays, Paul, Axiomatic Set Theory, Dover Publications 1991 Jacket: 'Since the beginning of the 2oth century, set theory ... has become increasingly important in almost all areas of mathematics and logic. In Part I of this excellent monograph, A A Fraenkl presents an introduction to the original Zermelo-Fraenkel form of set-theoretical axiomatics abd a history of its subsequent development. In Part II Paul Bernays offers an independent presentation of a formal system of axiomatic set theory, covering such topics as the frame of logic and class theory, general set theory, transfinite recursion, completing axioms, cardinal arithmetic, and strengthening of the axiom system.' Amazon back |
| 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.' Amazon back |
| Hallett, Michael, Cantorian set theory and limitation of size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson. Amazon back |
| 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.' Amazon back |
Click on an "Amazon" link in the booklist at the foot of the page to buy the book, see more details or search for similar items
Related sites:
Concordat Watch
Revealing Vatican attempts to propagate its religion by international treaty
