Some infinities are bigger than others

I just watched Fault in our Stars and of course started searching for the title of this post on Google. First a beautiful quote from this movie (Spoiler alert, if you haven't watched the movie).

In Hazel’s voice, Green writes,

“There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities.… I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful.”

This quote actually touches me personally.....We'll leave that philosophy :)

But guess what, mathematically she is wrong. Yes Cantor proved that some infinities are bigger than other's but it was not the infinities between 0 and 1 or 0 and 2 or 0 and a million (infinities between all three of these are actually equal).

It was for example, infinities between counting numbers compared with real numbers that are unequal. Check the following link on science360.com. It has an awesome description. You may want to watch the video below first before reading the link on science360.com.

Cantor's Infinity Proof on science360.com

Someone might question why I am gathering what's already available on internet on my own blog? Well, because I don't want to forget this. Anytime, I browse my own blog, it's going to remind me of Cantor's infinity proof which I intend to teach my kid as soon as he is ready to understand (my elder one is still only 5).