Monday, April 25, 2005

Golden Number and Different Infinities

Recently in my class on Transition to higher mathematics Dr. Jerzy Kocik has been giving lecture on Golden Numbers and its really fascinating. I liked his wooden divisor with three legs that divides any piece into golden ratio. The history behind it is interesting and the wonderful thing is its relation to Fibonacci numbers. Even though I know fibonacci numbers for now quite a while. It is this lecture by Dr. Kocik that helped me appreciate these numbers more fully. Another thing he talked about was Infinities which too was pretty neat. The numbers can be classified in two ways. Ordinal numbers and Cardinal numbers. Ordinal numbers have position or ranking. Cardinal numbers measure amount. And it is this concept of cardinal numbers that helps us define different infinities. Now the concept of natural numbers is one of the central one in number theory. Natural numbers are ordinal and integers are not. We say that the cardinality of natural numbers is aleph. I love the way this letter is written and Dr. Kocik's way of writing it. Now to count anything else we use bijection mapping. So in this way we can show that integers have the same cardinality as natural numbers. Even numbers, odd number and even rational numbers also have the same cardinality as natural number ie aleph. Here is how the numbers are defined in terms of set theory Zero is {}. One is {{}}. Two is {{},{{}}} and three you can guess is {{},{{}}, {{},{{}}} } and so on. This is the way ordinal numbers are defined. And also leads to the famous property "Well ordering principle". Which say natural numbers are well ordered. The cardinal numbers are based on aleph concept. Which is a power set concept. That alephZero (Yes there are alephOne, alephTwo and so on thanks to Cantor) is equal to Power set of Natural numbers. Then size of real numbers is equal to alephOne as one can show that there are more irrational numbers than rational numbers and there you see those cantors diagrams of counting. Indeed its fascinating to learn about infinity and zero and thats a huge revealation that there are levels of infinites. Bigger infinities. Cool kind of stuff we engineers don't get to learn.
While browsing through wikipedia math website I came across the concept of Bertrand russels objection and the barber paradox which reads like "There is a town in which there is a male barber who shaves every men who doesn't shave himself and no one else". Such a town couldn't exist. This paradox is called as Russell's Paradox.
Also I came to know that there are two kinds of set theory. One is called "Naive set theory" and other is called "Axiomatic set theory". The Naive set theory is based on set as a collection of elements whereas the "Axiomatic set theory" is based on axioms. The Goodel showed there are still discrepancies in the theory of sets. So far it remains a field open to more research.


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