Big and small infinities

This week involved examining infinite sets. The professor started with a "simple" infinite set, the set of all natural numbers. This set is easy to understand, all numbers that you can count starting from one. It is a simple concept and it is easy to see how one could get to extremely large numbers and still be able to continue. Therefore this is an infinite set because it can continue forever without ending. Next the professor proposed a simple idea. What about the set of all even numbers? It has the same idea as the set of natural numbers, just count each even number starting from 0 or even 2. This set is also infinite yet somehow it is half as big as the infinite set of natural numbers. When thinking of even numbers it is easy to think of them as half of the numbers, since even numbers appear on every second integer. I found this thought interesting, that although the set of natural numbers is infinite, the set of even numbers (which is also infinite) is somehow smaller. That there are smaller and larger infinities. The last set we looked at was the set of rational numbers between 0 and 1 which is even more infinite! despite being a small interval in the natural numbers, the set contains even more fractions than the set of natural numbers. Firstly, no matter how many decimal places you have listed, it is perfectly reasonable to add another decimal place an infinite amount of times. Secondly, it is always possible to add another decimal that is not already in the set.
I gained some insight from another students blogs about how infinite sets relate to python. Ji Yong Choi, related the infinite set to the number of possible programs that could be written in python. He also mentioned that most of these programs probably wouldn't work, or could cause an infinite loop. I found it discouraging to think one is more likely to write a program that doesn't work than one that does.