How are some infinities bigger than others?

Question

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

Answer 1

>I was confused as to why you can’t do this with natural numbers

Because then they’re not natural numbers. Natural numbers *must* be integers. Let’s try “adding one to the first digit of the first number and one to the second digit of the second number etc”:

* 1 is a natural number.
* 2.1 is not
* 3.01 is not

The Natural Numbers are 0, 1, 2, 3… But there are infinite numbers between 0 and 1. And another infinite numbers between 1 and 2, and another between 2 and 3…

The smaller infinities are called “**countable**” and the bigger ones “**uncountable**”. Here’s why:

1. Count the natural numbers: OK. 0,1,2,3,4…. It would take forever to count them all. But *if you had forever, you could do it*!
2. Now, count the numbers between 0 and 1, decimal numbers allowed: Alright. 0 ….um, *what’s the next number*? Is it 0.1? No, can’t be, then you’d have missed 0.01. Is it 0.01? No then you’d have missed 0.001…

**If you try and count the numbers between 0 and 1, you can’t even get to the first number past 0 in the list**, because no matter what number you say next, you’ll have missed an *infinite* number of numbers before that one! When trying to list the numbers between 0 and 1, there’s infinities *inside* infinities. An infinite number of numbers between each of an infinite number of numbers. It’s **uncountable**.

Listing the natural numbers doesn’t have this paradoxical-explosion-of-infinities problem. There’s just a “regular” list of numbers. It’s infinitely long, but each step is well defined and not growing within itself.