The Infinite

The idea of infinity is one of the most fascinating concepts in mathematics. I could talk for pages and pages about it, but I am going to focus on few fun and interesting infinity facts.
Everybody has an intuitive idea of what a set is. You could say this is my set of fine china, and the only things that would belong to that set would be the things you consider your fine china. There are an infinite number of examples for sets and I want to look at a couple interesting ones.
The set of the natural number (1,2,3,4,5,....) is an easy set to grasp; it's just the counting numbers going on forever. It is easy to see that there are an infinite number of numbers in this set. Now lets look at another set, the set of the natural numbers squared (1,4,9,16,25,...). Now, this too is obviously an infinite set, the numbers continue on forever. So, would you say that these two sets are the same size, i.e. they have the same number of numbers, or is one bigger than the other? Some people would say that they are clearly different sizes because the second set is missing all kinds of numbers (2,3,5,6,7,8,10,11,...) are all numbers that are not a part of the second set and the first set contains all of these numbers, and all of the numbers in the second set, so there must be more numbers in the first set.
I am going to claim that there are the same exact number of elements in the first set as in the second set, and I will prove it to you right now.

Let's try to match up the elements from the first set to elements of the second set. How will we do this? Well an easy way is to just match the number from the first set to its square in the second set.

1-1
2-4
3-9
4-16
5-25
.
.
.
And so on.
So you can see that if we continued this forever, there will always be a number in the second set for each and every number in the first set; it's square to be exact. And if there is an exact matching from one set into another, than those two sets are the same size.
I wanted to talk about this example because it shows what kinds of crazy things can happen when you move from the finite to the infinite. Things become quite bizarre and unintuitive, but very interesting.