I was very excited to look at the work Georg Cantor did for math in class. This was fueled by several reasons. For instance, Georg Cantor was extremely controversial with his discovery of multiple infinities. Also, I read about him in Concepts of Modern Mathematics by Ian Stewart and I’m doing my final project on the guy. For the sake of time and lengthy explanation needed to explain countable infinities and uncountable infinities I am going to assume that you already have an understanding of these concepts.
So as I previously stated, I am doing my final project on Georg Cantor and his work on infinity. I have looked into the Continuum Hypothesis, which says “There is not a set whose cardinality is strictly between that of the integers and the real numbers”, with the understanding that most mathematicians believe that they hypothesis is true, false, or impossible to determine. It is always fun to have people disagree, which is probably why I took such great interest in Cantor’s work. But in reality when researching and understanding these two separate infinities, and their abstractness altogether, I thought of it as fun, but pointless stuff. I don’t mean to try to take away the weight of the remarkable fact that there is such a thing as countable infinity, but I saw no significant reason in this being a huge discovery.
Well I was wrong in my thinking because after reading Concepts of Modern Mathematics I found that infinite cardinals were the key to proving the existence of transcendental numbers and how they compare to algebraic numbers. (A algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. A transcendental number is just the opposite. So the square root of 2 is an algebraic number, but pi and e are both transcendental numbers.)
Cantor proves this first by taking the height of a polynomial. It should be noted that there is a finite amount of polynomials that can be created with any specific height given. Cantor goes on to show that for an arbitrary polynomial, any whole number m we define f(m) to be the (m+1)-th distinct algebraic number along the sequence. This makes f a function N to the set of algebraic numbers. He proves that f is a bijection, which means the set of algebraic numbers has cardinal aleph-zero. Cantor knew that there were real numbers in the algebraic set such as the square root of 2 and he knew that the real numbers form an uncountable set. So this proved that there were some real numbers that are not algebraic, thus proving the existence of transcendental numbers. It also showed that there are more transcendental numbers than algebraic numbers. Suppose there were aleph-zero transcendental numbers, which would mean that
aleph-zero (algebraic) + aleph-zero (transcendental) = c(cardinal of R)
That is not true because aleph-zero plus aleph-zero equals aleph-zero. This was revolutionary because mathematicians had believed that transcendental numbers as being very rare and this proved that there was significantly many more than algebraic. This showed that almost all real numbers were transcendental. *Mind blown*
I am surprised that I did not come across this with my research on the internet. I found this gold mine by reading a book that just so happen to touch on Georg Cantor’s work. I think that the Continuum Hypothesis has casted a large shadow over his work on proving transcendental and algebraic numbers unfortunately.