How to count infinity.
Today, we're going to count Infinity. Now counting may seem elementary, like when we say we have 5 sheep we really mean we have 1 sheep for every number from 1 to 5, and 10 means we have 1 for every number from 1 to 10. So we say that 2 sets have the same number of things in them, simply if you can draw a line relating every item in one set to something in the other, and vice versa, exactly once. Their partners. It's the
same when we say that 2 + 1 = 3 or 3 doesn't equal 4, we're just describing the lines you can draw to relate one set of things to another, but either way counting sheep is boring, that is, unless you want to count infinitely many sheep, like if you had a sheep for everyone number between 0 and 2 would that be more sheep than everyone number between 0 and 1? Nope. Because you can relate every number between 0 and 1 to it's double giving you every number between 0 and 2 and if you want to undo, you can just divide every number between 0 and 2 in half to get back all the numbers between 0 and 1. But there are more real numbers between 0 and 1 then there are in the infinite set of integers 1, 2, 3, 4... and so on,
how on earth do we know this? Just draw some lines. For 1 draw a line to a number between 0 and 1 and for 2 draw another line to a number between 0 and 1. For 3 draw a line to a number between... 0 and 1, and so on, but no matter what numbers between 0 and 1 we've drawn lines to we can always right down a number between 0 and 1 that disagrees with the first digit here, the second digit here and the third digit here and so on, so this new number will be different from all the other numbers we've drawn lines to. But we've already drawn a line from every integer so there's no one let to be this numbers partner. Whats more, we can find an extra lonely number like this, no matter what numbers we've picked, which means we can never draws lines from the integers to all the numbers between 0 and 1 and this means there really are more numbers between
0 and 1 then there are in the already infinite set of counting numbers 1, 2, 3, 4... and so on. So some infinities truly are bigger then other infinities.