I'll tell you about some big, big things

Some bigger than others,

When one gets up, goes out the room,

He gets replaced by another.

Some infinities are bigger than others.

You have the countable infinities: the natural numbers (1,2 and so on) is one. They are infinite because they plod on, linearly, just being one big number after another, like a line of bullies forever meeting someone bigger than them (God's at the end of the line).

Then you have the uncountable infinities: the irrational numbers (can't be expressed as the ratio of two numbers) is one. They are called irrational numbers as, for example, exactly unlike 2 is the ratio between 6 and 3 (and 8 and 4 and so on), the square root of 2 is not the ratio of any other two numbers. So the square root of 2 is an irrational number.

You'd think these would be few and far between. They're not; they're many and near between: every couple of rational numbers has an infinite amount of irrational numbers between them.

In fact there's so many irrational numbers you can't count them. You can try, for every one of the pug-faced bullies in that never-ending line, assign an irrational number. You'd think that the bully assigned the square root of two would be fairly close to the beginning of the queue. He wouldn't, he'd be so far up the line, he'd have gone beyond the end. And it doesn't have an end.

(as it's infinite, which is kind of the point of this, so if you're lost now, don't worry, the paragraph coming up is roughly halfway up the page, so if you look to the right of that (no, your right), you'll see a list of posts where I swear a lot)

And I can(tor) prove it.

First some preliminaries:

How to recognise an irrational number, cook it and eat it

(more on this later)