I was watching Veritasium’s video about math having a fatal flaw. He explained that if we make a set of all the numbers in between 0 and 1, then added one to the first digit of the first number and added one to the second digit of the second number etc, we would always have a new number. He said this proved that there were more numbers in between 0 and 1 than natural numbers.
I was confused as to why you can’t do this with natural numbers, and how that proved one infinity was smaller than another.
In: Mathematics
>… Why you can’t do this with natural numbers
Well, in order to do the trick, you need an ordered list first. You need something that clearly comes after another thing, so that you can assing an ‘ID’ to a natural number the same way Vertitasium uses a natural number as an ID for the real ones.
But the thing is, what else is there but natural numbers? Nothing else is ‘naturally ordered’. You could use the alphabet, but there is no reason why it should be ordered the way it is in the first place. A being first is arbitrary.
But 1 coming before 2 isn’t, and it’s the only thing we can all agree on. It’s also the reason why it’s called a countable infinity.
> He explained that if we make a set of all the numbers in between 0 and 1
Not just a set, a list. It has to be well-ordered for the proof to work. There has to be a first number and a second number and so on. They can’t all be in a disorganized heap.
Remember, the point of the proof is to show that such a list is *impossible.* The argument is, no matter what list you present, his method will find a Real number 0 < x < 1 that isn’t on the list.
The thing about math is, in some ways, it’s like a game. The rules are whatever we say they are. Math is a human invention so all of its rules and definitions are whatever humans have come up with.
So we have decided that, in math, we can have things called sets. Sets are just like collections of things. And sets can have sizes, which we define by the number of things in that set.
But what about the size of sets that have an endless amount of things? Like the set of all natural numbers? Or the set of all real numbers?
Well, in this case we have decided that to sets with an endless amount of things are the same size if you can create a rule that maps every single object in each of the sets on a 1-to-1 basis without missing any in either set.
For example, take the natural numbers {0, 1, 2, 3….}
Then take the even numbers {0, 2, 4, 6…}
Clearly there are “more” natural numbers than even numbers, right? Because the natural numbers *include* the even numbers, plus more!
Wrong.
Given our rule, they are the same size because we can map them 1-to-1. Take any natural number, multiply it by 2 and you get a unique even number. Taken any even number, divide it by 2 and you get a unique natural number. 1-to-1 mapping. They are the **same size.**
So we began to wonder if all sets with an endless amount of objects were the same size.
Back in the day, a dude named Cantor came up with a rather elegant argument that showed that the set of real numbers is actually bigger than the set of natural numbers. He created a proof that showed that, no matter what rule you created to map the natural numbers to the real numbers, that there would exist real numbers not accounted for in that mapping. That there would always be and endless amount of real numbers left over.
For that reason, we consider the set of real numbers as being “larger” than the set of natural numbers.
Notice that, at no point have I used the word infinity. Unfortunately, “endless” *means* “infinite” and both of those sets we were talking about (natural and real numbers) are endless, so they are both infinite, yet according to our rules of math, the latter is larger than the former. Hence, “some infinities are bigger than others.”
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>I was confused as to why you can’t do this with natural numbers,
If you take a natural number, and change one of its digits after the decimal place, you don’t get another natural number. If you start with, say, 3, and change the ten thousandths place, you get 3.0001, which isn’t a natural number— so it’s not a problem that it’s not found on your list of all the natural numbers.
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