I have not read the John Green novel but while watching The Fault in Our Stars I recognised the fault in the main character saying there is a bigger infinite set of numbers between 0 and 2 than 0 and 1.
I wrote some notes about infinity and then discovered I am by no means the first to point out the error in a blog entry. This entry http://thenumberghost.wordpress.com/2014/05/09/the-fault-in-our-math/ for instance has a groovy explanation of bijection functions which I needed as I had not heard of them.
Any way The Fault in Our Stars mentions infinity a few times and in particular Cantor.
Georg Cantor was a German mathematician best known as the inventor of set theory. Wikipedia tells me he proved that real numbers are “more numerous” than the natural numbers. His theorem implies the existence of an “infinity of infinities”.
I had heard of Cantor only in relation to him being one of the first investigators of fractals. The Cantor Set is an uncountable subset of R. The Cantor Set is a fractal, although it predates the existence of that word.
Wikipedia entry on cardinal number includes:
Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either m or n is infinite, then
m + n = max{m,n}
Therefore the film in saying that there are an infinite number of numbers between 0 and 1 and a bigger infinite number of numbers between 0 and 2 is either wrong or has something interesting to say about the axiom of choice.
There is a distinction between the cardinal number of the set of natural numbers, which is deemed a countable infinity, and uncountable infinities. It is counter-intuitive to think that any infinity is termed countable!
What is meant by saying one has unlimited choice?
Infinite choice or boundless choices for a biological system must pose a problem. There simply isn’t time to consider an endless number of outcomes. I think what we mean when we say a person has unfettered choice of action is that there is such a large number of options that we can not list them all yet the total number is finite despite being very large. The person considers groupings of similar actions without enormous detail for every single case. There are after all only so many ways that matter at our disposal can be rearranged, which also negates the idea of infinite choice. The restriction of reality on the number of choices that can be actually enacted does not extend to impossible actions. The number of imaginary actions probably is infinite. I think even then those that could be considered by a human are restricted by the physical limitations of the brain.
We only have a lifetime to formulate ideas so when making up imaginary impossible actions there is a still a restriction of experience in actuality. Perhaps on the edge of imagination there are acts of creative thought that bring new conceptions into existence that previously were impossible. I guess as I believe there are new things under the sun (as opposed to “There is nothing new under the sun”) I side with the emergence of new ideas that break from the past.
Cantor himself said that some of his ideas were revealed to him by God. It is now believed that Cantor had bipolar. Cantor hated infinitesimals. My take is that the world is a rich but finite place but it has an ever growing number of poor in it.
matthew simpson 