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
First, we need to agree on what a list is. A list is just a way to assign a real number to each integer. There are a lot of possible lists, so just to make things clear, here is a possible list with an obvious rule of construction :
* 1) 0.1
* 2) 0.11
* 3) 0.111
* 4) 0.1111
* 5) 0.11111
* …
And so on
This is a particular list of real numbers between 0 and 1. It contains infinitely many items (one for each integer) but it certainly does not contain all the real numbers between 0 and 1. For exemple, it does not contain 0.2
Given any list of real numbers, there is a procedure (known as diagonalisation) that will return a real number that is not on that specific list. To illustrate the procedure, let’s just take an example with a beginning of a list
* 1) 0.**1**45869735…
* 2) 0.2**3**3813848…
* 3) 0.12**1**565351…
* 4) 0.835**8**35833…
* 5) 0.1535**7**3831…
* …
Take the digits in bold (**1** **3** **1** **8** **7** …) , add 1 to each digit (2 4 2 9 8 …) , and create a new number by concatenating these new digits after a “0.” . This gives will give a number starting with : 0.24298…
(Note that this procedure always gives a number which is a real number between 0 and 1)
This new number cannot be anywhere on the list. Indeed, for any integer n, the n-th digit of that new number is different from the n-th digit of the n-th number on that list.
Now, one can answer the following question : “Can we make a list of real numbers that contains all real numbers between 0 and 1 ?” The answer is NO. Because whatever the list is, there is always at least one real number that is not on that list, which is the one created by the procedure.
This happens because there are far too many real numbers. Which means that even if you try to make an infinite list of real numbers, you will always miss some of them. So we translate that by saying that the infinity of real numbers is larger than the infinity of integers.
You can not do the same procedure with a list of integers, because the object created by the procedure of concatenating digits is not an integer.
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