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• Voiceover: Any sequence you can come up with,

• whatever pattern looks fun.

• followed by random numbers, whatever.

• All of that plus every sequence you can't come up with,

• each of those are the decimal places

• of a badly named so called real number

• and any of those sequences

• with one random digit changed

• is another real number.

• That's the thing most people don't realize

• about the set of all real numbers.

• It includes every possible combination of digits

• extending infinitely among aleph null decimal places.

• There's no last digit.

• The number of digits is greater than any real number,

• any counting number

• which makes it an infinite number of digits.

• Just barely an infinite number of digits

• because it's only barely greater

• than any finite number

• but even though it's only the smallest possible

• infinity of digits.

• This infinity is still no joke.

• It's still big enough, that for example

• point nine repeating is exactly precisely one

• and not epsilon less.

• You don't get that kind of point nine repeating

• equals one action

• unless your infinity really is infinite.

• You may have heard that some infinities

• are bigger than other infinities.

• This is metaphorically resonant and all

• but whether infinity really exists

• or if anything can last forever

• or whether a life contains infinite moments.

• Those aren't the kind of questions

• you can answer with math

• but if life does contain infinite moments,

• one for each real number time,

• that you can do math to.

• This time, we're not just going to do metaphors.

• We're going to prove it.

• Understanding different infinities

• starts with some really basic questions

• like is five bigger than four.

• You learned that it is

• but how do you know?

• Because this many is more than this many,

• they're both just one hand equal to each other

• except to fold it into slightly different shapes.

• Unless you're already abstracting out

• the idea of numbers and how you learn

• they're suppose to work

• just as you learned a long life

• is supposed to be somehow more than a short life

• rather than just a life equal to any other

• but folded into a different shape.

• Yeah, metaphorically resonate that.

• Is five and six bigger than 12?

• Five and six is two things after all

• and twelve is just one thing and what about infinity?

• If I want to make up a number bigger than infinity,

• how would I know whether it really is bigger

• and not just the same infinity

• folded into a different shape?

• The way five plus five

• is just another shape for 10.

• One way to make a big number

• is to take a number of numbers, meta numbers.

• This is where a box containing five and six

• has two things and is actually bigger than a box

• with only the number 12.

• You could take the number of numbers from one to five

• and put them in a box

• and you'd have a box set of five

• or you could take the number of numbers

• that are five which is one

• or you could take the number of counting numbers

• or the number of real numbers.

• It's kind of funny that the number of counting numbers

• is not itself a counting number

• but an infinite number often referred to

• as aleph null.

• This size of infinity is usually called countable infinity

• because it's like counting infinitely

• but I like James Grime's way of calling it

• listable infinity because the usual counting numbers

• basically make an infinite list

• and many other numbers of numbers are also listable.

• You can put all positive whole numbers

• on an infinite list like this.

• You can put all whole numbers including negative ones

• by alternating.

• You can list all whole numbers

• along with all half way points between them.

• You can even list all the rational numbers

• by cleverly going through all possible combinations

• of one whole number divided by another whole number.

• All countably infinite numbers of things,

• all aleph null.

• Countable infinity is like saying

• if I make an infinite list of these things,

• I can list all the things.

• The weird thing is that it seems like this definition

• should be obvious that no matter how many things there are,

• of course you can list all of them.

• If your list is literally infinite

• but nope so back to the reals.

• Say you want to list all the real numbers.

• If you did, it could start something like this

• but the specifics don't matter

• because we're about to prove

• that there's too many real numbers to fit

• even on an infinite list

• no matter how clever you are at list finding.

• What matters is the idea that you can create

• any real number you want,

• out of an infinite sequence of digits

• and we're going to use this power to create a number

• that couldn't possibly be on the list

• no matter what the list is

• even though the list is infinite.

• All we need to do that is construct a real number

• that isn't the first number on the list

• and isn't the second number on the list

• and isn't any number on the list

• no matter what the list is.

• Here's where I'm sure some of you are like "Yes!"

• Cantor's diagonal proof.

• Indeed my friends, that's what's going down.

• In the first number on the list,

• the first digit is one.

• If I make a new number with a first digit is three

• then even the rest of the digits are the same,

• there's no way my new number

• is equal to the first number on the list

• though the rest of the digits

• probably aren't all the same anyway.

• The second number on the list does start with a three.

• We don't know if this new number is the same or not yet

• but I can make sure my new constructed number

• is not the second number on the list

• by making the second digit five

• or eight or whatever

• and I can make my number not be the third number

• on the list

• by making the third digit five

• I mean the new number was already different

• from the third number on the list

• but I don't even have to check the other digits

• as long as I know that one of them

• definitely conflicts which comes in handy

• when I get to the 20 billion and oneth

• number on the list

• and I don't have to check the first 20 billion digits

• against the 20 billion digits

• I've constructed so far to be sure

• that my new number is not the same

• as the 20 billion and oneth number.

• There's one digit in my number

• for every number on the list

• which means I can make a way for my new number

• to not match every single number on the list

• no matter what the list is.

• Which means there's more real numbers than fit

• on an infinite list.

• This works no matter what the list is.

• Take the diagonal and add two to every digit

• or add five or whatever.

• You can't actually sit down and write an infinite list

• or infinite number though.

• Here's another way to think about what's really going on.

• We're trying to create a function that maps

• one set of numbers to another.

• You can map all the counting numbers