So, on the previous page, we compared the set of natural numbers to many different infinite sets, and found that there was a bijection between the natural numbers and most infinite sets except for the real numbers. This is because there are actually two different sizes of infinity: countable (listable) and uncountable (unlistable). We will see just what we mean by that in this video.
Did you figure out why the real numbers are unlistable? I will go over it again just in case.
Instead of looking at the entire real number line, let us first simply look at the line segment from 0 to 1. Let's list some of the real numbers from 0 to 1:
- 0.098765421...
- 0.187894533...
- 1.000000000...
- 0.000200051...
- 0.134897128...
- 0.172384932...
Ok. The reason for this proof is to mind a "missing" number from this list, no matter how gargantuan the list gets. The point is there will always be a missing number. You have to look at the diagonal. What do I mean by this? Consider the following same list, but with some numbers bolded.
- 0.098765421...
- 0.187894533...
- 1.000000000...
- 0.000200051...
- 0.134897128...
- 0.172384932...
You take the first decimal digit of the first number, the second decimal digit of the second number, the third of the third, the fourth of the fourth, and so on and so on. If you combine them, you will get the number 0.080294... Now, all you have to do is change every decimal digit somehow. Of course, there are many different ways of doing this. James Grime in the video changed every 1 to a 2, and every other number to a 1. This way, every single decimal digit is different. I prefer to add 1 to every digit, and not carrying any ones, just changing the 9s to 0s. Of course, there is no "right" way of doing this, so feel free to make up your own formula to change the digits.
Anyways, you come up with a completely different number than any number on that list. It can't be the first number, because the first decimal digit of our "missing" number is different. It's not the second number because the second decimal digit is different too. It can't be the seventeen millionth number because the seventeen millionth digit of our number is different. So we have found an entirely new number not on this list at all. And you can do this for every list of numbers anyone gives you, no matter how long it is. That is why the real numbers are considered "unlistable".
Extra: (read this only if you want even more confusion to fill your brain)
For every list of real numbers, you can actually find infinite many different missing numbers by using infinite different methods of changing decimal digits. For instance, if we used James Grime's method, we would get a different missing number than if we just added 1 to each decimal digit, or subtracted 1 from each digit, or multiplied 7 to each decimal digit and don't carry, except if there is a 5 or a 0, in which case you add one. You can find some completely ridiculuous methods, but that still create unique missing numbers.Reveal Confusion